In GIS, vector and raster are two different ways of representing spatial data. However, the distinction between vector and raster data types is not unique to GIS: here is an example from the graphic design world which might be clearer.
Raster data is made up of pixels (or cells), and each pixel has an associated value. Simplifying slightly, a digital photograph is an example of a raster dataset where each pixel value corresponds to a particular colour. In GIS, the pixel values may represent elevation above sea level, or chemical concentrations, or rainfall etc. The key point is that all of this data is represented as a grid of (usually square) cells. The difference between a digital elevation model (DEM) in GIS and a digital photograph is that the DEM includes additional information describing where the edges of the image are located in the real world, together with how big each cell is on the ground. This means that your GIS can position your raster images (DEM, hillshade, slope map etc.) correctly relative to one another, and this allows you to build up your map.
Vector data consists of individual points, which (for 2D data) are stored as pairs of (x, y) co-ordinates. The points may be joined in a particular order to create lines, or joined into closed rings to create polygons, but all vector data fundamentally consists of lists of co-ordinates that define vertices, together with rules to determine whether and how those vertices are joined.
Note that whereas raster data consists of an array of regularly spaced cells, the points in a vector dataset need not be regularly spaced.
In many cases, both vector and raster representations of the same data are possible:
At this scale, there is very little difference between the vector representation and the "fine" (small pixel size) raster representation. However, if you zoomed in closely, you'd see the polygon edges of the fine raster would start to become pixelated, whereas the vector representation would remain crisp. In the "coarse" raster the pixelation is already clearly visible, even at this scale.
Vector and raster datasets have different strengths and weaknesses, some of which are described in the thread linked to by @wetland. When performing GIS analysis, it's important to think about the most appropriate data format for your needs. In particular, careful use of raster algebra can often produce results much, much faster than the equivalent vector workflow.