Play basics. The Finesse
Playing low towards high
The three fundamentals of general play are:
- Play low towards high
- Attack your length
- Keep winners, throw away losers
It is the first fundamental which is most frequently overlooked by novices. Failure to play low towards high accounts for half the errors a new player makes.
Most of us have heard the advice to play "2nd player low" and "third player high". For example:
The opening leader leads the 2
. Dummy plays the 3 (2nd player low). The next defender plays the King. . .. third player high, to force out Declarer's Ace. Notice that both pairs have played a low card towards a high one (2 to King, 3 to A). The maxims about playing "2nd player low" and "3rd player high" are simply applications of the basic guideline "play low towards high".
Take this common situation
Dummy 
K 5
Declarer 4 2
If a Club is played from Dummy we will never take a trick. The defence will simply beat the King whenever it is played to them. But, if Declarer leads from his hand and plays low towards Dummy's high card (i.e. the King) we will make a trick whenever the Ace is in the West hand.
Now revise the hand slightly
Dummy A Q
Declarer 4 3
Here we will take one trick with the Ace but might get a second trick if we come to our hand and lead a Club towards Dummy, playing the Queen (if and only if West has the King, a 50% chance). We have, in effect, increased our chances of taking two Club tricks from 0% to 50%. This simple plan is called a "finesse" but, like "2nd player low" and "3rd player high", is nothing more than another application of our simple "play low towards high" rule.
This same plan pops up again when we have a holding such as:
Dummy AQ10
Declarer 32
One Club trick is all we get if we play from Dummy. But if we lead towards our big Clubs in dummy and insert the 10 (on the first round) and then again towards the Queen we might win three tricks in the suit if West (LHO) has the King and the Jack (a 25% chance)! Because we are hoping West has two cards (i.e. King and Jack here) this is called a "double finesse".
We occasionally have to lose a trick in a suit in order to finesse later.
Dummy A J 10
Declarer 4 3
Once more, playing the Ace from Dummy will win us only one trick. But if we lead towards the 10 and, when that loses to the King or Queen, repeat this finesse we can make two Club tricks whenever West has the King and/or the Queen (a 75% chance). Note that we can only make these two tricks if we lead towards our big cards (i.e. Dummy's AJ10) twice here, finessing each time.
Playing the 10 on the first round here is called a "deep finesse".
There are two corollaries to this basic "play low towards high" tenet.
a) Start long towards short
When we have high cards in both hands it's best to play towards the shorter hand first. This helps unblock the suit so that we can run it later. It also helps us to finesse in the suit later if an opponent has no cards in this suit, or "shows out".
Dummy 
Q 2
Declarer A K 10 4 3
If we play the Ace or King on the first round we will block the suit. If we play the Q first we can run the suit whenever the J falls doubleton or tripleton. If West shows out on the first round we can finesse for the J on the second round.
Dummy 
Q 2
West K J 8. East 9 7 6
Declarer A 10 5 4 3
Again, we start with a Diamond from the long hand towards our Q2. If West plays the King our troubles are over when the enemy holding splits 3-3. If West ducks the Diamond, though, we will still win with the Queen. So, when the suit breaks 3-3 we will take four Diamond tricks.
b) Open towards hidden
Dummy K 10 4 3 2
West J 9 8. East A 7 6 5
Declarer Q
If Declarer were to lead the Queen here East would usually take it with the Ace. But if we play from the "open" (i.e. visible) hand, Dummy, East might duck and allow our Queen to win this trick.
c) When to first play a top card
Dummy 9 4 3 2
West Q. East 10 8 7 6
Declarer A K J 5
Here you have two top cards, the A and K. The finesse can still be played if one of them has been used. So play the Ace first, in case the Queen is bare, as in this example. Otherwise you will lose this finesse unnecessarily. There's nothing to lose, and by discovering the distribution (e.g. if one player shows out) you can sometimes adjust the play to your advantage.
But beware playing a top card from the long hand if that should create a blockage, see (a) above.
A "Sandwich" Finesse
This describes the situation where the missing honour card is sandwiched by honours in your two hands, AND you have at least two honours below the missing card. By playing this finesse, the extra card below the missing card becomes promoted.
In the following example the King is missing, but you can make 2 tricks if the King is with West.
Dummy A 5
West Kxxxx. East 7xxx
Declarer Q J
The method. Lead from the WORSE holding, leading the Queen. If the Queen holds, play the 5. If it doesn't hold, cover it with the Ace. The Jack is now promoted.
In the following example, leading the Jack has the effect of promoting the 10 if it's covered by the Queen then King: if it's not covered the Jack wins anyway.
Dummy A K10 2
West Qxx. East 9xxx
Declarer J 3
(Good defence will play the Queen, so that at least the 4th trick wins with the promoted 9).
When not to finesse
When . . .
- it's more likely that the missing card will drop (see 1006),
- you have a singleton (in trump contracts),
- you know (from the bidding or the play) that the finesse will lose,
- losing the finesse will put the lead into dangerous enemy hands, leading to certain trick loss.