The easiest way to get your head around vector as opposed to raster is to understand how a computer reads the file formats differently.
All graphic files are rendered from left to right, line by line.
Imagine you have a grid of 100 x 100 pixels. Top left corner being 0 x 0, bottom right corner being 99 x 99.
Raster files will (generally) process by reading pixel 0 x 0 = red, 0 x 1 = green etc. JPG (and some other raster file formats) will bunch coordinated blocks of pixels together. 0 x 1 to 0 x 36 = red. This equates to a smaller file size and faster processing/rendering, but some loss of detail.
Vector files use equations which will adapt to a percentage rather than pixel x pixel. Over simplifying, but a vector file will measure the coordinates of the vector making for a much smaller file size if the graphic is a simple line drawing. 0 x 0 to 0 x 20 + 1 x 1 to 2 x 20 + 3 x 20 = black. Rather than the computer having to process pixel by pixel, a single equation will render a greater percentage of the image. For vector think 'percentages'. For raster think in 'pixels'.
A pixel is a finite thing - a vector, or 'percentage' is infinite. You can't stretch pixels without harming them. Vectors will stretch forever!
If you want to create, or output images with complicated graduations, or photographs stick with raster formats. For line drawings and simple graphics, go with vectors. That said, there are some very impressive vectors artists creating incredibly complicated illustrations.
If you want to have a simple play at converting one to the other, get yourself an old copy of Macromedia Flash - an animation tool highly underrated for its graphics ability. Import a photograph and play with 'Trace Bitmap'. You will soon get a good understanding of the difference between each basic format.
There are also hybrid formats which will render both as vector and raster, but they aren't really worth exploring unless you are seriously concerned about processing limitations.