On what basis is each chromosome given a number?
Chromosomes are generally ordered by size in terms of number of nucleotides. The first chromosome is the biggest. Note that when there are sexual chromosomes (as it is the case in mammals or birds for example), the pair of sexual chromosomes are typically considered last irrespective of their sizes.

Note that in some species the ordering has been decided and further studies have shown that the actual size order does not perfectly match the ordering. This is for example the case in humans, where chromosome 11 is actually bigger than chromosome 10 (thanks @canadianer).

Numbering is mainly just a simple naming system. It is just handy to be able to refer to a chromosome as chromosome 3 rather than having to somehow describe this chromosome for other to understand what we are talking about.