Heart Finesse is really two words that should be dealt with separately. Heart is just one of the four suits in a standard deck of cards. One could substitute Clubs, Diamonds or Spades and use it in the same phrase.
Finesse is the more interesting term. It is a form of a dilemma posed to one's opponents in a game of bridge.
When a team makes a contract in bridge they are obligating themselves to capture a certain number of tricks. For this example we'll choose a contact of "Four Spades". This obligates the team to take 10 tricks. The total number of tricks available is 13 and a bid requires (6+the bid) to be successful.
Absent trumps, high cards win each trick in bridge, so aces, kings and queens take most tricks.
To continue this example, the player who made trump is going to realize that he can only make 9 tricks with some certainly and needs to figure out how to gain an extra trick. A common way to do this is to use a finesse.
The finesse utilizes a gap in the owner's suit of cards and tries to force an opponent into not getting a trick for a high card.
To continue our example assume that the declarer has AQ in hearts and 72 on dummy. After capturing a trick on dummy he leads the 7. The opponent now has a dilemma, she can play the King only to see it fall to the Ace, or she can play a lower card and Declarer will cover it with his Queen. In this way, the Declarer can possibly get two heart tricks when only one was assured.
This is an example of a direct finesse. There are many other kinds of finesses which you can read about at Wikipedia. They all have the same basic theme though, trying to get an extra trick (or more) by exploiting a gap in a sequence of cards.
Without any special knowledge a finesse has a basic 50% chance of success. You can only finesse one opponent and the card you are trying to finesse could always be with the other opponent. Usually, due to the bidding nature of Bridge you have a pretty good guess at which opponent to finesse, but you never quite get to 100% certainty!
Depending on your knowledge of cards and Bridge in particular I don't doubt that I used terms that are unfamiliar. Please ask clarifying questions or leave comments and I will explain further.