I'm barely beginning to play again, but I would have to define the following as:
Check: A pokemon that will come in not-so-easily, maybe after death fodder and threaten to out-beat the opposing pokemon whether immediatly or proceedingly. An example could be switching in a Swampert on a Charizard after it kills some other pokemon. Charizard can still do heavy damage, but you're opponent is swayed into switching because it will eventually outbeat it.
Counter: A pokemon that outright threatens the opposing pokemon with resistances, blocking strategies, set up windows and of course an easy KO.
An example of this could be a Blissey switching in on a DisableGar. Of course he can thunderbolt or focus blast all he wants but Blissey is using the opportunity to heal and set up. The more offensive side would be Choice Band Scizor switching in on a Mamoswine or Terrakion.
Both are meant to switch momentum in battle, and this is what I have understood from the two terms. Keep in mind you can use a counter to { v.check} another Pokemon
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