The idiot end of a straight refers to a straight where the three of four high cards come from the community cards.

Given the above cards in a game of hold'em, a player with 8♣ 9♦ would have a straight, but it'd be the idiot end of the straight. This is said because it is quite possible that another player could have AK and thus have a higher straight.

The expression is typically used when a player with such a straight is very aggressive and ignores the possibility of being beaten by a higher valued straight.

Such a situation is even more pronounced on a board like the following.

A player with a nine has a straight. Though many starting hands in hold'em contain an ace, and there are an abundance of ace-anything players, thus an ace-high straight is very likely in this situation. Furthermore, that ace-high straight is the nuts at this point. The player with the nine has no chance of getting that high straight to fold.