Ok, this is for beginners, so I've left in certain things which I tend to agree confuses matters, such as the result. I will be doing analysis as best I can first with open cards, then with closed cards to illustrate the dangers of attributing any value to the result. First - the hand.

$0.05/$0.1 No Limit Holdem
10 players
Converted at weaktight.com

Stacks:
UTG ($9.90)
UTG+1 ($1.90)
UTG+2 ($4.70)
MP1 ($10.12)
MP2 ($6.02)
Hero ($12.44)
CO ($8.45)
BTN ($42.34)
SB ($10.39)
BB ($4.55)

Pre-flop: ($0.15, 10 players) Hero is MP3
5 folds, Hero raises to $0.4, 1 fold, BTN calls $0.4, 1 fold, BB calls $0.3

Flop: ($1.25, 3 players)
BB checks, Hero bets $0.8, BTN calls $0.8, BB folds

Turn: ($2.85, 2 players)
Hero bets $1.9, BTN calls $1.9

River: ($6.65, 2 players)
Hero checks, BTN checks

Final Pot: $6.65
BTN shows:
Hero shows:

BTN wins $6.32 ( won +$3.22 )
Hero lost -$3.10
BB lost -$0.40

I had 50 or so hands history on the villain with his stats around 25/12/2 and I hadn't seen him in any showdowns - and besides I've only recently begun taking hand reads seriously so wouldn't have any notes on him anyway.

PF thinking:
I see a QJo, which I'd possibly rather not play 2 positions off the button, but the table has been folding an awful lot and I'd decided to open my range anyway. On this table I judge there's a 50% or better chance of me taking down the blinds with it and I can see a few flops that it could hit hard while also being good for other hands so I'm not too worried about opening with it. I get two callers, one of them on the button so I'm OOP for the hand. Not the best news.

Playing with PokerStove:
I will go through the hand first with the pocket cards face-up. For a beginner (like me) I think it is a useful counterpoint to the hand range analysis that we are limited to as we play the hand.

PokerStove rates the Equity PF thus:
QcJs: 38.89%
AhTh: 61.11%

PokerStove rates the Equity on the flop thus:
QcJs: 74.34%
AhTh: 25.66%

PokerStove rates the Equity on the turn thus:
QcJs: 88.64%
AhTh: 11.36%

It is tempting as a beginner to go through a superficial analysis like this (the actual cards do matter, right?) and come to the conclusion that in this case the Hero did nothing wrong. It is also wrong.

When you look through a Hand History of a hand that went to showdown you are not trying to determine if you did the right thing in the hand based on the actual cards - (it is done using ranges instead - see below) - but rather trying to determine how the villain plays in this specific spot so you can take good notes on him. A note I'm considering for this villain is this:
Doesn't 3bet ATs on the BTN

I don't think I can fault him for calling the near 2/3 bet on the flop as he has a gutshot. The flop is scary, but it would be perfectly reasonable for him to think I cbet any missed flop. I don't think it is necessarily wrong of him not to raise in this position on the flop. So no comment on that account.

On the turn his hand improves. In the face of a second barrel he's calling with top and bottom two and a gutshot. With the second barrel he can't really assume that he's ahead unless he's up against total air, and he has 0, 2, 6 or 9 outs to improve his hand to a winning hand (4 J to the straight which beats any trips or two pair, 3 A which beats any two pair, 2 T to a boat which beats any two pair, straight or trips or 0 outs if the Hero is holding KJ or KT - yes K could make a boat also, but wouldn't help if Hero has K or J - more complications). The PokerStove calculations can be done from his perspective also, putting me on ranges etc to determine if he did the right thing.

And then he checks the boat on the river. I'm not sure that's a check-behind situation for him. I think he should have value bet here.

Going back to PokerStove - now using ranges instead of his actual cards. I'll put in a 13% range, adjust it to 13% slightly towards lower PPs, suited connectors (and away from Axs) just assume he'll PF 3bet AA, KK, QQ, AKs, AKo and AQs and deduct those from the 13% range.

PokerStove rates the Equity PF thus:
QcJs: 42.30%
JJ-22,AJs-ATs,KJs+,QJs,JTs,T9s,98s,87s,AQo-AJo,KQo: 57.70%

(Deducting no hands to give Equity straight on the flop before betting)
PokerStove rates the Equity on the flop thus:
QcJs: 74.71%
JJ-22,AJs-ATs,KJs+,QJs,JTs,T9s,98s,87s,AQo-AJo,KQo: 25.29%

Now - we have to deduct hands. We have to deduct the hands that would have raised the flop bet and the ones that would have folded to the flop bet.
Deducting 22-TT, 87s-98s (leaving in T9s due to the straight draw)
The only hands I think should raise the flop are AJ and QJ as they have top two and are worried about drawing hands.
PokerStove rates the Equity on the turn thus:
QcJs: 52.56%
JJ,ATs,KJs+,JTs,T9s,AQo,KQo: 47.44%

Now, I should have known on the turn that my equity had fallenfrom around 75% to 53% simply because he called. But I didn't think of that - and that may make my turn bet questionable. In a couple of iterations of this analysis I left in AJ and QJ and my equity fell to 32% or so. Including AJo in his range is also not too far fetched and would also skew the numbers. My equity on the turn is 53% and not 89% because the hand he happened to hold is not the only hand he would have played the exact same way.

Actual thought process:
On the flop I'm very scared of the KK and I'm perfectly happy if I can take down the pot right here. I have top two and it's not a bad hand, but lots of Ks and drawing hands are in (both) my villains' ranges. I lead in with a 2/3 pot bet trying to appear unruffled by the flop and hope they will put me on a K and fold.
On the turn.. I probably tilt. I see that my hand improves! I now have an open ended straight draw. I can't lose! I forget to consider my opponents range - I particularly forget to consider that all the hands I can beat just folded to me, but this guy didn't. I'm still slightly worried about straight draws and kings, but with my straight draw to counter trips I think I'm probably much better off and happily fire away at the turn.

EDIT:
River thinking which I forgot initially.

With the T pairing it dawns on me how much of his range I'm now behind. Any T or K would have me beat. AQ a very probable hand. I decide to slow down and call a smallish value bet only - $2-3 or so.
END EDIT

When the hand ended I thought I had a great hand to take to analysis and sat out of my table to come here and share it.

To those who consider the above somewhat pedestrian I have two questions:
1) Is this on the whole a sound analysis? (and further to this point) Is a 13% range reasonable for his PF call with his stats? It is around his PFR (12%) and below his vp$ip (25%). I know it's a judgement call, but do people generally call a standard raise only with hands they would PFR, more hands or fewer hands?
2) Based on this particular Hand History which notes would you take on the Hero and on the Villain?