Can anyone sum up when to use these lines on the river in a nutshell? Lately I've been getting into a lot of these situations and it'd be nice to hear opinions. Specific examples would be real nice.

Let's start with this one:

$0.25/$0.5 No Limit Holdem
6 players
Converted at

Stacks:
UTG ($37.75)
UTG+1 ($49.75)
CO ($24.55)
BTN ($19.55)
SB ($62.25)
Hero ($63.55)

Pre-flop: ($0.75, 6 players) Hero is BB [Qd]
UTG calls $0.5, 4 folds, Hero raises to $2.25, UTG calls $1.75

Flop: [Jc][7h] ($4.75, 2 players)
Hero bets $3, UTG calls $3

Turn: ($10.75, 2 players)
Hero checks, UTG checks

River: ($10.75, 2 players)
Hero bets $6.5, UTG calls $6.5

Final Pot: $23.75

Bet/Fold when you feel your opponents range is weighted towards made hands and you feel you are ahead on average.

Check/Call when you feel your opponents range is weighted toward busted draws or at other times when you think a check may induce a bluff from a mid strength hand.

bet/call is the new check/call imo