That's a broad question, but some distinctions are: - 'cross' can be a verb as in 'cross the street', 'cross the river' 'across' is an adverb as in 'we could see across the street', 'we paddled across the river' 'through' usually means going into and out of something, as in 'the train went through a tunnel in the mountain' - but it can also be used figuratively to mean 'examine' as in 'I looked through his draft essay', but that's less common.