How is the GT field in a VCF file defined?
0 and 1 is just a way of coding the reference (0) and alternate (1) allele. These could be A/G, C/T etc. It's just a simplified way of expressing the different alleles.
0/1 and 1/0 functionally mean the same thing (that the individual is a heterozygote) - since the genotype is unphased, the alleles aren't ordered. The / symbol tells you the genotype is unphased. However, in practice, a heterozygous genotype is always written as 0/1 as a matter of convention. 0/0 is referred to as homozygous reference, and 1/1 as homozygous alternative.
0|1 and 1|0 do mean something different, since the pipe symbol | tell us the order of the alleles matters.
Occasionally you will have multi allelic positions, where you have more than one alternate allele, and therefore the field will look like:
Ref   Alt
A   [C,GT]

The extra alt allele is given the number 2, so in this case if an individual had the genotype CT then their genotype code would be 0/2.
Full detail is given in the official VCF specification page.