Perpetual check is when a player can check the other infinite times in a row, with no reasonable chance of escape. In this situation, the game will eventually be a draw if the player giving the series of checks does not let up, either by the fifty move rule or by threefold repetition if the players don't mutually agree a draw before one of these cases happens.
In the strictest case, perpetual check can refer to those positions where there is absolutely no way for the defender to escape the series of checks, even by making a material or positional sacrifice. However, in a large number of cases that are also normally referred to as perpetual check as well, there are one or more ways for the defender to escape the series, but they lead to a significant enough disadvantage as to make doing so not viable from a game standpoint (i.e., refusing the draw would lead to a loss).
Two examples will suffice. In Diagram 1, Black is significantly ahead on material (he has a bishop for a mere pawn in exchange) and White would certainly lose the game if it continued in any normal way (Black will simply make way for his queen to defend along the second rank if White tries to attack by, say, 1 Rg3, and White hasn't enough force to achieve anything). However, White can force a draw by perpetual check by a sacrifice of his queen:
1 Qxh7+!! Kxh7 2 Rh3+ Kg6 3 Rg3+ etc.
And White simply checks again and again at g3 and h3, producing a draw. This is an example of the "strict" form of perpetual check, as Black has no way of escaping the series of checks at all. (In fact, if he tries to get away by moving his king to f5, White will even win, because his Rook will have been at g3 allowing him to play the check Rg5+, which is actually mate!)
In Diagram 2, it is not as urgent for White to seek the shelter of a draw. Black, nonetheless, appears to have the advantage, due to a stronger situation in the center combined with the fact that White's c2 pawn is a backward pawn that is likely to become weak, especially since Black can fairly easily exchange White's d3-bishop (via, say, Kh8 followed by Bf5. Not of course Bf5?? right away, because White would simply play Bxf5 and Black's g6-pawn, being pinned will not be able to recapture!) and in many tournament situations White would look for a draw. And in matter of fact he does have a draw by perpetual check, again with a sacrifice to remove the pawn cover from Black's king:
1 Bxg6! hxg6 2 Qxg6+ Kh8 3 Qh6+ Kg8 4 Qg6+ etc.
This is an example of the technical (i.e., there is a way out but it's a losing one) type of perpetual check. There are two ways for Black to get out of the checks, but in both cases he loses his queen - in fact, both ways because of a pin: Nh7 in reply to Qh6+ loses the queen because once White's queen arrives at the sixth rank, it pins the Black knight to the unguarded queen at d6, and if Black tries to run away from the checks by moving his king to f8 instead of h8 in reply to Qg6+, White plays Ba3, and this time it's Black's queen that gets pinned - and lost. Therefore, Black has no choice but to accept the draw, because his only other options involve definitely losing the game.