This page usually contains a piece of theory, in this case about 'to finesse or not to finesse', including an example.
Three exercises will follow, one in each of the next three weeks (click 'Varia***' ).

Both examples below are taken from team matches with IMP-scoring, so declarer's priority is making the contract; overtricks are relatively unimportant.

S/-		♠	3 2
		♥	A K Q
		♦	K J 7 5 4 3
		♣	5 3
		♠	K J 4
		♥	7 6 2
		♦	A 2
		♣	A 10 8 4 2

¹ NS play this as forcing, 10⁺ points (some pairs prefer 8-11, non-forcing)

West leads the ♠6, via East's ♠9 to declarer's ♠J.

South needs at least four diamond tricks. His best chance not to lose a diamond trick is finessing the ♦Q. If he decides to do so, he plays the ♦A, then the ♦2 to the ♦J.
But if East wins and (of course) plays a spade back, declarer is defeated (unless the spade suit is 4-4). East is therefore the danger hand.
There is a better way to play the diamond suit. Declarer can afford to lose a diamond trick (in principle even two) as long as he keeps East off-lead. If he loses a diamond trick to West, there is no problem (if West played back a spade South would make a second trick in that suit).
If East has three ore more diamonds including the ♦Q, declarer does not stand a chance. He can guard against the ♦Q being the doubleton in East though, by cashing the ♦AK, followed by a third diamond: if West turns out to have three diamonds to the ♦Q, declarer has has given up no more than an overtrick.
However, by playing diamonds from the top South would still risk unnecessary defeat:

S/-		♠	3 2
		♥	A K Q
		♦	K J 7 5 4 3
		♣	5 3
♠	A Q 10 8 6			♠	9 7 5
♥	J 9 3			♥	10 8 5 4
♦	Q			♦	10 9 8 6
♣	K 9 7 6			♣	Q J
		♠	K J 4
		♥	7 6 2
		♦	A 2
		♣	A 10 8 4 2

On this lie of the cards East will eventually make a diamond trick: down one.
Declarer can do better still, because he can guard against this layout as well. At the second trick he does not play the ♦A but starts off with the ♦2. On seeing West's ♦Q, declarer ducks in dummy*. This way he makes certain he loses the inescapable diamond loser to West.
The rest is simple: declarer wins West's return, unblocks the ♦A and crosses to a high heart for the remainder of the diamonds: ten tricks.

If (different layout) at the first diamond trick West follows suit with a small card, declarer wins with the ♦K and plays a diamond to the ♦A. This way he guards against the ♦Q doubleton (or bare) with East.
If the ♦Q has not come down, declarer crosses to dummy with a heart and plays a third round of diamonds. He now has done all he can to keep East off-lead and must hope for his last chance: West having started with three diamonds to the ♦Q.
(True: if West has started with four diamonds to the ♦Q, the 'standard play' of the ♦A, followed by the finesse over West would succeed, unlike the recommended play of a small diamond from South first. However, in view of West's length in spades he is less likely to have length in diamonds. Besides, if West has that unlikely four card diamond suit to the queen, declarer still has a chance: he may be able to strip off West's hearts and clubs to enplay west to lead back spades).

* A clever West player will therefore play that card if he has the ♦Q doubleton, saving an overtrick!

Sometimes theory tells us not to finesse. For instance if declarer and dummy have nine cards in the combined hands and the queen is missing. However, that theory is valid only when looking at that specific suit in isolation: in order to take the best chance of not losing a trick to the queen, the best play is indeed to lay down the ace and king then.
But in dummy play declarer can have many reasons to finesse after all. The bidding for instance, or the distribution of the other suits. After all, a well designed plan is based upon the whole deal, not just one suit.

S/All		♠	J 10 5 4
		♥	A K Q 10
		♦	7 5 2
		♣	A 6
		♠	A K 9 7 6
		♥	J 7 2
		♦	K 4 3
		♣	K 2

¹ What else? A direct 4♠-bid would show a weak, distributional hand (this problem is one of the reasons top players bid a conventional 2NT bid: forcing, spade fit; this bid is called 'Truscott without the double' in most of Europe, Jacoby in the US, Stenberg in Norway)
² Since South has shown a balanced 12-14 hand, North simply settles for game; a forcing 3♠ bid is now pointless

West leads the ♣Q. South counts four possible losers: three diamonds (if the ♦A is wrong) and the ♠Q.
He wins in hand and lays down the ♠A, EW contributing small spades.
It looks like this is a matter of catching the ♠Q. The 'normal' play seems to be to play the ♠K since this offers the best chance of not losing a trick to the ♠Q. If an opponent shows up with the guarded ♠Q, declarer can hope for him to have at least three hearts, so South can pitch a diamond in time. And as a last chance the ♦A might be with East, right?
On closer inspection: that line of play even wins if West turns up with the guarded ♠Q: after the ♠K (East showing out) declarer plays four rounds of hearts, discarding a diamond. West is welcome to ruff whenever he wants, since he cannot do any harm: if he returns a diamond South is happy to make the ♦K (and an overtrick) and if he plays a club dummy's ♣A serves as the entry for the remaining heart(s).
So declarer lays down the ♠K and then plays the hearts?

S/-		♠	J 10 5 4
		♥	A K Q 10
		♦	7 5 2
		♣	A 6
♠	3			♠	Q 8 2
♥	8 6 5 3			♥	9 4
♦	A 9 8			♦	Q J 10 6
♣	Q J 10 9 8			♣	7 5 4 3
		♠	A K 9 7 6
		♥	J 7 2
		♦	K 4 3
		♣	K 2

Unlucky, very unlucky: East has the guarded ♠Q, ruffs the third heart, plays the ♦Q and of course the ♦A is wrong! Down one.
On closer inspection: bad luck had nothing to do with it! Declarer could have done better still, because after EW had both followed suit to the ♠A, the contract was ironclad.
Declarer should realise EW can only take three diamond tricks if East gets the lead and plays a diamond. Declarer must be careful therefore not to let East in. After playing the ♠A he crosses with a heart and plays the ♠J from dummy. If East shows out or plays the ♠Q (on a different layout from the one shown), South wins with the ♠A and plays off the hearts; no problem, as mentioned before.
But if East produces a small spade, South finesses. If West shows out, like here, South is happy: he draws the ♠Q with the ♠A and starts on the hearts. He ends up with eleven (if the ♦A is wrong) or twelve tricks.
But what if the finesse fails, West winning with the ♠Q? Well, South has only lost an overtrick. The contract is safe since he can always pitch a diamond on the fourth heart.
Thanks to this line of play declarer never loses more than three tricks: the ♠Q and two diamonds.