Which piece is powerful? Bishop or Knight?

On the battlefield knights are much more powerful with their lance or longsword. Most bishops only have an unwieldy crosier, and are quite old and feeble.

However, bishops are much more influential, given their empowerment to confer holy orders.

Every single move by both sides changes the value of all the bishops and all the knights. Maybe by only a small margin but there is a change. One of your knights may be more valuable than the other. One of your bishops may be less valuable than your knights or your opponents knights.

In the starting position the bishops are slightly more valuable than the knights and after that the value of the knights and bishops change [if ever so slightly]

Sometimes combinations of pieces change evaluations. In the opening position a Q and B is worth more than a Q and N.  However if you get to an end game with only some pawns and one side has Q and B and the other side has Q and N--then in that case the Q and N are more powerful.

The simple answer is: they're roughtly equal. 

The more complex answer is:

While bishops are arguably slightly more powerful given the fact that the bishop pair confers a slight (roughly .5 pawn) advantage to the owner, the specifics of a given position determine which piece is more powerful. Bishops prefer open boards, and are weakened if their own pawns are on squares of their color. They are further weakened if those pawns are immobile central pawns. A bishop's strength lay in it's activity, so even a "bad" bishop can outshine a knight if the bishop is actively placed and serving a specific positional demand. 

Knights, by contrast flourish on relatively closed boards. In order to exercise their full power knights need outpost squares. They are considerably weakened if the position does not offer such squares, or if the only squares are on immaterial parts of the board. 

Additionally, it matters what major pieces are left as well. Bishops tend to be more powerful if there is at least one rook of the same color left on the board, while knights tend to shine when combined with a queen. 

Alot of the relative value of bishop versus knight depends, therefore on which major pieces are left, how many pawns remain on the board, what color squares the fixed pawns occupy, how open or closed the position is, how actively the bishop can be placed, and what outpost squares are available for the knight. 

But over millions of games, these differences even out, and we know that on average, over those many, many games, the bishop and knight are about of equal worth, in general. But in any specific position one will be better than the other, and it is in being able to accurately discern which is better in a specific position (and by extension, how that position should be most accurately played) that chess players demonstrate their skill. 

Probably the most improtant thing to consider in evaluating the relative strength of knights and bishops is the pawn structure.

There are ways that players can evaluate these things and these have been at least partially given by some of the postings here.

Watson makes a good point in Secret of Modern Chess strategy on this topic. What he says it whilst it is true that in theory bishops are long range pieces and therefore better in open positions, it doesn't follow from that that the player with the two bishops should always try to open the position up straight away.

Rationale: in practice, the side gaining the two bishops usually does so at the expense of some other concession: a pawn weakness, an initiative to the opponent, etc. Therefore, by opening the position up, he can allow his opponent to exploit said advantage. It's often better for the side with the bishop pair to take stock first, work out what weaknesses there are in his/her position, and work to neutralise them first, before opening the position and activation the bishops at the right moment, when then can use their long-range power most effectively.

thanks for the tip...

Depends on the position, the way the game develops and also they way you like to play.

I'd say keep every piece active (and you king safe) and they will all have equal value.

If I had a choice, I would take White in this position.

Of course it is totally contrived.