What's the difference between a projection and a datum?
Geographic coordinate systems (lat/long) are based on a spheroidal (either truly spherical or ellipsoidal) surface that approximates the surface of the earth. A datum typically defines the surface (ex radius for a sphere, major axis and minor axis or inverse flattening for an ellipsoid) and the position of the surface relative to the center of the earth. An example of a datum is NAD 1927, described below
Ellipsoid Semimajor axis† Semiminor axis† Inverse flattening†† Clarke 1866 6378206.4 m 6356583.8 m 294.978698214
All coordinates are referenced to a datum (even if it is unknown). If you see data in a geographic coordinate system, such as GCS_North_American_1927, it is unprojected and is in Lat/Long, and in this case, referenced to the NAD 1927 datum.
A Projection is a series of transformations which convert the location of points on a curved surface(the reference surface or datum) to locations on flat plane (ie transforms coordinates from one coordinate reference system to another).
The datum is a integral part of the projection, as projected coordinated systems are based on geographic coordinates, which are in turn referenced to a datum. It is possible, and even common for datasets to be in the same projection, but be referenced to different datums, and therefore have different coordinate values. For example, the State Plane coordinate systems can be referenced to either NAD83 and NAD27 datums. The transformations from geographic to projected coordinates are the same, but as the geographic coordinates are different depending on the datum, the resulting projected coordinates will also be different.
Also, projecting data may result in a datum conversion as well, for example, projecting NAD_1927 data to Web Mercator will require a datum shift to WGS 84. Similarly, it is possible to convert data from one datum to another without projecting it, as with the NGS's NADCON utility, which can shift coordinates from NAD27 to NAD83.
Example of a point's coordinates referenced to different datums
Coordinates referenced to NAD_1927_CGQ77
19.048667 26.666038 Decimal Degrees Spheroid: Clarke_1866 Semimajor Axis: 6378206.4000000004 Semiminor Axis: 6356583.7999989809
Same point referenced to NAD_1983_CSRS
19.048248 26.666876 Decimal Degrees Spheroid: GRS_1980 Semimajor Axis: 6378137.0000000000 Semiminor Axis: 6356752.3141403561
After struggling with this question ten years ago, and finding many confusing things written about the topic, I published a brief article in Directions Magazine that presented an answer as simply, plainly, and accurately as I could make it. The following is excerpted from that article.
Reprojecting geographic features
Two things must happen when you draw a map: features in the real world must be "georeferenced" to a spheroid and the spheroid must be projected onto the paper.
The spheroid models the shape of the earth's surface. It is an idealization that does not account for local changes in topography.
Georeferencing assigns locations (in three dimensions!) to points on a spheroid.
Projecting is an operation that mathematically distorts and shrinks a portion of the spheroid onto flat paper. Projecting can be undone ("inverted"). "Unprojection" expands a feature on a map and plasters it back onto the spheroid.It, too, is a mathematical operation.
Georeferencing is done with a datum. A datum is usually given by a starting point and direction: it specifies where a clearly identifiable point on earth (the base point) should appear on the spheroid and it shows where a base direction, such as north, points on the spheroid at the base point. The base point and direction allow surveyors to determine the distance and angle of any other point on the earth. Moving in the corresponding direction on the spheroid for the same distance determines where the new point should go on the spheroid.
Spheroids have coordinates. They are latitude and longitude. (Geodetic) latitude is the angle made by a vertical line to the horizontal. It is not necessarily the same angle made by "straight up," because the latter is distorted by gravitational variation over the earth. It is not necessarily the angle made by a line to the center of the earth, because most spheroids have an elliptical cross-section, not a circular one.
Therefore, georeferencing endows points near the earth with latitude, longitude, and height coordinates.
(Subsequent sections discuss Change of datum, How to relate two maps, The wrong way to do it, and North America is a special case.)
wwnick's answer is correct, but it is a bit misleading in the sense that it emphasizes ellipsoid parameters and IMO understates the importance of 'the position of the surface relative to the center of the earth' - the NAD 1927 example needs to mention that the geodetic "center" of NAD27 is a base station at Meades Ranch in Kansas.
One could have (and often that's the case, especially with the increasing popularity of WGS84/GRS80 ellipsoid) several different datums based on the exact same ellipsoid parameters. The reason for this is that while the WGS 84 datum is OK globally since its surface is set to provide minimal average shifts due to tectonic movements across the globe, there's room for improvement on the local scale, where the reference can be fixed to some local reference point or at least to the local tectonic plate (e.g. ETRS, which is fixed to continental Europe)
One could explain datum simply as "an agreement on the coordinate system type, shape and its absolute position and orientation relative to some well-known or well-defined real-world reference". The coordinate system doesn't even have to be ellipsoidal (e.g. Vertical datum, which is usually defined by saying that the height of some fixed point is such, and all other heights will be measured relative to this point).
Geographic projections are a way of showing the curved surface of the Earth on a flat surface like a piece of paper...
From the Manifold user documentation:
Earth is not an exact ellipsoid. In fact, because the Earth is such a "lumpy" ellipsoid no single smooth ellipsoid will provide a perfect reference surface for the entire Earth. The practical solution to this is to measure the Earth's shape in different areas and to then create different reference ellipsoids used for mapping different regions on Earth. A datum is a reference ellipsoid together with an offset from the center of the Earth. By specifying different offsets, you can use the same standard ellipsoids in many different regions of the Earth. Different countries will often use the same ellipsoid but with different offsets for standard government maps in those countries.
Think of projection as seeing your location on X/Y plane. Datum defines the reference point from where all measurements were made. Say you are located somewhere and need to tell your location to someone. You would say, i am X lat and Y long. This X and Y are deterministic because they are being referred from the Datum. The other person now knows that you are X-lat and Y-Long away from Datum. If you are a newbie, dont concentrate too much on Datum characteristics. Just remember that its the location from where all measurements are made.
I wrote an in-depth article on this on my blog here: http://www.sharpgis.net/post/2007/05/05/Spatial-references2c-coordinate-systems2c-projections2c-datums2c-ellipsoids-e28093-confusing
It covers all these concepts in a hopefully easy to understand manner, and has been peer-reviewed by several.
To sum it up: A datum is a definition of the size, orientation and position of an ellipsoid used as an approximation of the earths shape. It uses reference points on the surface to define it's placement and orientation, based on a date (which is why a number is in there for the year it was defined to account for tectonic plate movements). Datums are used in both spherical long/lat and projected coordinate systems. Consider it a reference point for your coordinates and ellipsoidal heights (ie where's the primemeridian, equator, and what's the height relative to the ellipsoid which isn't the mean sea level). Different datums are used different places because some fit some areas better than others.
A projection is a formula used to convert long/lat coordinates into a flat coordinate system that you can use on paper or a computer screen. It's usually done from a geographic coordinate system, which in turn uses a datum as it's base definition. So the datum affects all of it. Projecting data creates a lot of distortion of the real world, so it really should only be done when putting your map data on a flat map, or you want to work in a "simpler" coordinate system and can live with the distortions.
Using the wrong datum could result in your data being offset up to about a mile, so it's quite important to know the datum if you're mixing data together.
This won't compete with wwnicks answer and not rigorous, but the visualization I present to people, when asked, is the relationship between a string connected to a ball. Changing the projection is often like moving the 'loose' end of the string around, but still connected to the same point on the ball. Changing the datum is like changing the location of the ball. This might help those visual types.
We should remember the earth is not a simple sphere, if it was, we need one datum "= One calculation system to find a point on earth", earth is more ellipsoid, but not exactly. Earth is an astronomic geoid without a regular shape, so we may have many ways to calculate coordination of a point in this irregular 3D object, with many opinions and concepts, each one is a datum.
ICSM's Fundamentals of Mapping page on Datums 1 – The Basics can be visited for more information.
In short, a projection is used to 'flatten' the ellipsoidal shape of the earth to rectangular coordinate system (e.g., a map). A datum is a specific, known point on or in the Earth that is used for reference. A projection uses the datum as a point of reference, it's location on Earth.
In GIS, there are two types of "coordinate systems": Geographic Coordinate System (latitude and longitude) and Projected Coordinate System (X and Y). Both geographic coordinate system's and projected coordinate systems use a datum for reference.
A geographic coordinate system is not projected (not flat), they are in latitude and longitude. Think of a round globe, not a flat map.
Projected coordinate systems on the other hand are "flat" - but still need a point of reference (datum) to define locations in space.
In other words, the datum is used to determine the point of origin on Earth by referencing a central point inside a 'model' of Earth.
Just a comment on the diagram that is trying to illustrate a projection from a sphere. Rather that what is illustrated, imagine a light source at the center of the sphere. The shadow of the polygon "projected" onto a flat piece of paper outside of the sphere is in essence a type of projection. To me the diagram is implying a projection is like a reflected surface which is an incorrect way to visualize what is happening.
Also, at least in the ESRI world, georeferencing is not applying points to a sphere. Georeferencing is assigning a known planar (projected) coordinate system to either a raster or vector dataset that has originated from either a scanning or digitizing operation in which a 'local' coordinate system was first applied. "Local" in this case simply means the coordinates were made up with no reference to a real world coordinate system. That is, a map may have originally been hand digitized where the person decided the lower left coordinate of the map had an XY value of (0,0). Georeferencing is the process of assigning a set of real world (projected) coordinates to the original. If this process is applied to a photograph or scanned map then the georeferencing process will often warp the original image to fit within the set of reference points that have been assigned real world planar coordinates. This "georeference warping" is not the same as the distortions created when projecting from a sphere onto a plane. "Georeference warping" is all about correcting distortions produced by either the camera or scanner. When projecting a feature from a spherical surface to a planar surface there is always a distortion created in distance, area, scale, and bearing. You choose a projection to minimize one or more of these distortions, depending on the intended purpose of the map.
As to the strings on a ball illustration and changing the datum, rather than strings I would use pencils of various lengths that start from a point on the sphere and end on a flat piece of paper. The outer ends of the pencils represent the projected points. In a sense, changing the geographic coordinate system (datum for this discussion) is analogous to rotating the sphere on one more axis to a new position. The concept works only for isolated areas on the earth. That is for NAD27 to WGS84 it applies pretty well to the 48 contiguous states of the USA but not for Canada or Alaska. For those areas you have to correct the NAD 27 datum first and then make the NAD7 to WGS84 move. Whereas for NAD83 to WGS84 the concept works for most of North America.