Image by pashabo.
The first and most important feature is their ultimate scalability - the ability to be resized without losing quality, as opposed to raster graphics, which can blur or pixelate excessively if enlarged too much. This property is due images being made of a set of mathematical curves, rather than separate pixels. This applies to fonts as well - most fonts of Postscript and TrueType format are in fact stored as vector graphics, which enables their scaling in most computer programs.
The second important thing about vector files is that their file size relies completely on the complexity of image, rather then its colour depth or size. Which means, that most of the time, the file size is smaller, than it's raster equivalents, making it good for sharing via internet. There are of course exceptions, images with complex techniques and effects (such as gradient mesh) and files with big number of anchors.
The third thing we would like to point out, is that vector files are easier to edit. Experienced graphic designers know, that sometimes re-making raster images is faster than trying to modify them, especially if we are talking about files, that had their layers merged. This is never the case with vector images - the lines, points and colour fill are always there, and can be changed easily with only a couple clicks of a mouse.
Fourth thing of note is that PDF format files can contain vector images and preserve their scaling property (i. e. without jagged edges appearing even under 800% magnification), that is, if they are converted into PDF correctly.
Speaking of fifth property, we would like to draw a conclusion, that the ultimate scalability of vector images also makes them ideal for printing - for example, the same logo can easily be printed on a t-shirt and then enlarged to be printed on a giant advertising poster.
The sixth fact that contributes largely to the vector image small sizes is that files only contain data about the Bézier curves and their control points which form the drawing. And final, seventh fact about vector graphics - the high precision and the best known scalability make vector graphics perfect for drawing schematics and blueprints; many software solutions exist for this, such as, for example, ConceptDraw.