While there are 19 species of snowdrop only a few are likely to be found in Britain (outside of specialist collections/botanic gardens). The commonest is definitely Galanthus nivalis which is almost the only one that you might find somewhere that looks truly wild (i.e not in, or next to an old garden or churchyard). This has a double flowered form "flore pleno" which is also very widely naturalised.
The main things to look at when trying to identify snowdrops are the way the leaves are arranged and their shape and colour (Grey-green, Green or Bluish-grey) and the pattern and extent of the green markings on the inner parts of the flower.
Leaves can emerge from the bulb with the two leaves flattish, facing each other and be the same size (applanate) *see picture - our common snowdrop (G. nivalis) is like this. A slightly different leaf form sees the margins of the leaf narrowly tucked under (explicate) - this type is shown by G. plicatus.
The other main type has one leaf base wrapping round the other (supervolute) *see picture - plants with bluish-grey leaves are likely to be Galanthus elwesii, those with glossy bright green leaves are likely to be G. woronowii (often mislabelled ikariae in garden centres).
Many of the bigger, stronger growing garden forms are actually hybrids, usually of G. nivalis with G.plicatus - they have broader leaves than nivalis (wider than a little finger-nail) and you can see that one or more of the leaf margins shows the narrowly folded over edge shown by plicatus. The more garden worthy double forms known as the Greatorex hybrids are of this parentage.
G. nivalis and most of the commonly grown species have a single green mark at the end of the inner flower segments but G. elwesii typically has two discrete green patches - one at the top, one at the bottom - although in a lot of plants these two merge together and many plants of G. elwesii in cultivation only have a single apical green patch (these are often incorrectly labelled G. caucasicus in garden centres) - they can be told from nivalis (and plicatus) by their leaf arrangement.
Identifying named cultivars, of which there are hundreds is not easy - some are very distinctive because they show strikingly abnormal flowers but many differ in only small subtle ways. To try to get to grips with these it is best to look in the excellent monograph by Grimshaw et al. where pictures and detailed descriptions are given - there are also snowdrop picture galleries on the web where you can compare your plants with named examples.