Raster graphics (such as in Photoshop, GIMP, etc.) are a grid of pixels. Their main problem is that they do not scale up and down very well, without losing a lot of information. These include GIF, JPEG, PNG, TIFF, and most other image formats. Vector graphics (e.g., SVG) use an arbitrary coordinate grid to draw lines between any two points, and then map it to the desired resolution and size of pixels without any loss of information. Vector graphics take a lot more work to fill an area or to do gradations of color or shade. Their chief advantage over raster (pixel-based) formats is that they scale up or down perfectly, as far as you want to go without permanent loss of detail, and can be noticeably smaller (file sizes). The chief disadvantage is that the rendering load is generally placed on the browser, and may have to be repeated each time the image is displayed.
Note that vector graphics are no good for photographs. They are better for diagrams, charts, etc., where you have well-defined lines on solid colors, possibly with well-defined gradients. If the user will be viewing diagrams and wishes to zoom in and out, a browser-based rendering of SVG may be the answer. If the diagram or chart data is frequently updated, vector graphics avoid having to regenerate a raster image on the server each time.
Also note that if you are looking at vector graphics to replace JPEG for diagrams and charts, because of the noticeable artifacts near color boundaries, that GIF may be better choice (lossless compression is used, which doesn't introduce compression artifacts). JPEG uses lossy compression, which tends to introduce artifacts due to the discarding of high-frequency information during compression.