If I have contour lines with interval say 10, 5, or 2 (etc.), and I generated a DEM from them, what is the relationship between between the contour interval and the resolution of the DEM? Irrespective of the software I am using, whether QGIS, ArcGIS, or any other software.
Contour interval is a single number that defines the number of vertical units between contours. You can't have 10, 5, 2 unless you have different sets of contours. If it's 5, each line represents 5 vertical feet of change between lines. You can of course highlight 10m contours on an interval of 2 (every fifth) or 5 (every other). Contour lines are an interpolation of continuous point samples (or a grid, which is what the DEM is).
There are two resolutions to a DEM - vertical and horizontal. Resolution is the minimum distinguishable distance between samples. Vertical is controlled by the precision of the elevation value stored in the cell, and mostly relates to when you're actually measuring or have a stated source. Horizontal you can control with the cell size in that each vertical measurement is assigned to an area. If you are creating a DEM from contours, you are also interpolating (estimating unknown values from known). In fact, since contours are typically produced via interpolation, you're re-interpolating data.
When you interpolate, you have the potential to introduce false accuracy into your data. See When dealing with rasters of varying resolutions should one resample to the highest or lowest resolution? for more explanation on how this is so. Let's say you have a contour interval of 10m and your contours are uniformly 100m apart. This means your known values (the contour lines) are no closer than 100m horizontally or 10m vertically. You can create a DEM with cells that are smaller than that or larger, and that store values to the nearest one meter or ten. But the finer/higher resolution version won't truly be any more accurate than your source measurements.
One could say there is no direct relationship between the two in that changing one will not necessarily affect the other. The other way to look at it is the smaller your contour interval, the higher the resolution you can set for the DEM without introducing false accuracy or precision. Ideally you would at most match the two, or use a lower resolution than your interval.