In the good old days, cartographers drew maps on pieces of paper. Printed maps were easier to read than hand-drawn ones, but for centuries maps were fixed representations of a changing physical world. If something changed or you made a mistake, you had to redraw the map; if you wanted it at a different scale, you had to redraw it; if you wanted a map limited to one sort of information and another with a different sort of information, for example, a physical map of hills and rivers as opposed to a road map, you had to draw two different maps.
In recent decades, computers have completely changed this. Now, instead of lines on paper, computers store the definition of the lines and not the lines themselves. For example, a road or river might be a straight line from point a to point b, then a curve of such-and-such an arc to point c, then another straight line to point d, and so forth. These definitions are called vectors. Once you have this, you can rescale as often as you like without changing your definitions or losing any information. You can also associate any data with any point or line, for example, the name of the road and who is responsible for maintaining it. But this is not fixed to any representation of that information on paper. In short, you separate the vectors and the informational data from the way it appears on paper.
If you want a hard-copy, you decide the scale, which lines and data you want, the size of the lines, typesize of any placename labels, etc. The process of converting a particular subset of your data at a particular scale is called rasterisation. The roadmaps and topographical maps you buy in the shops are now largely rasters produced from computer-stored vectors.
Of course, once you have geographical information stored on computers, you are no longer limited to paper output; you can also display it on a computer monitor, and you can make it publicly available over the web so anyone can look at any mapping information you choose to provide them with. You can either provide your vector data 'raw' in a vector format, or you can distribute a particular view of all or part of your data, rasterised to one of the standard computer bitmap formats, such as JPEG or PNG.
Although the term 'vector' is widely used throughout the computer graphics industry, vectors in the geographical context are often called features. A geographical feature normally has a geometry, specifying where it is (coordinates) and what it is (point, linestring, polygon), and attributes or properties, which contain the associated data describing the feature. Point features on maps are also often referred to as points of interest or POI.