I'll start off with noting basic factors we consider when we want to decide whether we want to make a river c/r or not. I'm not sure which should be consider "main" factors, but all are definitely relevant.

C/r with nut hands:
1. Is our opponent going to value bet thin and on a similar note is he passive or aggressive? There are a lot of opponents at higher stakes especially that vbet thin like it's there job. Obviously, if we have a nut hand and our opponent is going to bet a wider range than he will call/raise with, a c/r may be the decision you want to go with.
2. Is he the type of person who cant find a fold button when faced with a minraise? Most of the time money is shallow, so most river c/r's are close to min.
3. Have we been making a lot of river c/r's?

More fun is c/r as a bluff.

1. Same as nut hands #1.
2. Is your image tight
3. Is he capable of making big folds.
4. Does our line make any sense?

Possible river c/r discussion hands:

I have been playing very aggro and 0evg0 is making the light vbetting like his life depends on it. I have been c/r the river a good amount, clearly sometimes with the nuts and clearly sometimes with not. He is a very good hand reader. This hand is obviously a bad play by me but worth discussing.

POKERSTARS GAME #12269058978: HOLD'EM NO LIMIT ($2/$4) - 2007/09/25 - 19:42:27 (ET)
Table 'Wurm' 2-max Seat #1 is the button
Seat 1: 0evg0 ($1033.50 in chips)
Seat 2: heybude ($621.50 in chips)
0evg0: posts small blind $2
heybude: posts big blind $4
*** HOLE CARDS ***
Dealt to heybude [Kh Qc]
0evg0 said, "****kkk"
0evg0: raises $8 to $12
heybude: raises $28 to $40
0evg0: calls $28
*** FLOP *** [Td 3s Ad]
heybude: bets $52
0evg0 said, "cooler after cooler"
0evg0: calls $52
*** TURN *** [Td 3s Ad] [8c]
heybude: checks
0evg0: checks
*** RIVER *** [Td 3s Ad 8c] [7s]
heybude: checks
0evg0: bets $100
heybude: raises $208 to $308


POKERSTARS GAME #12268409962: HOLD'EM NO LIMIT ($2/$4) - 2007/09/25 - 19:05:03 (ET)
Table 'Electra IV' 2-max Seat #2 is the button
Seat 1: 0evg0 ($394 in chips)
Seat 2: heybude ($406 in chips)
heybude: posts small blind $2
0evg0: posts big blind $4
*** HOLE CARDS ***
Dealt to 0evg0 [9d 8c]
heybude: raises $8 to $12
0evg0: calls $8
*** FLOP *** [Kc 9h 2d]
0evg0: bets $16
heybude: calls $16
*** TURN *** [Kc 9h 2d] [4d]
0evg0: checks
heybude: bets $44
0evg0: calls $44
*** RIVER *** [Kc 9h 2d 4d] [9s]
0evg0: checks
heybude: bets $104
0evg0: raises $164 to $268

POKERSTARS GAME #12269563212: HOLD'EM NO LIMIT ($2/$4) - 2007/09/25 - 20:10:56 (ET)
Table 'Sculptor' 2-max Seat #2 is the button
Seat 1: heybude ($1705.50 in chips)
Seat 2: 0evg0 ($734 in chips)
0evg0: posts small blind $2
heybude: posts big blind $4
*** HOLE CARDS ***
Dealt to heybude [Kc 8c]
0evg0: raises $8 to $12
heybude: calls $8
*** FLOP *** [Kd Tc 7c]
heybude: checks
0evg0: checks
*** TURN *** [Kd Tc 7c] [4d]
heybude: checks
0evg0: bets $16
heybude: calls $16
*** RIVER *** [Kd Tc 7c 4d] [6c]
heybude: checks
0evg0: bets $44
heybude: raises $104 to $148

Another horrible one
POKERSTARS GAME #12268174825: HOLD'EM NO LIMIT ($2/$4) - 2007/09/25 - 18:51:19 (ET)
Table 'Sculptor' 2-max Seat #2 is the button
Seat 1: heybude ($868.50 in chips)
Seat 2: 0evg0 ($539 in chips)
0evg0: posts small blind $2
heybude: posts big blind $4
*** HOLE CARDS ***
Dealt to heybude [4c 5c]
0evg0: raises $8 to $12
heybude: raises $28 to $40
0evg0: calls $28
*** FLOP *** [Th Qc 9s]
heybude: bets $52 (yeah horrible lets just discount this)
0evg0: calls $52
*** TURN *** [Th Qc 9s] [Qh]
heybude: checks
0evg0: checks
*** RIVER *** [Th Qc 9s Qh] [9c]
heybude: checks
0evg0: bets $145
heybude: raises $290 to $435