To the left of each post you have the poster's profile. You'll see their join date, number of posts etc.
You'll also see
1. A green box or boxes
3. Thanked x times in x posts
4. Rep power
I'm going to attempt to explain all that.
There are two ways you can show someone your appreciation for a post you liked or that was useful to you. 1) You can click the Thank You link to say thanks and 2) You can click the star below their post and leave them "reputation".
The Thank You is a quick way of showing appreciation. In other forums people post just to say thank you. On these boards such a post would be considered fluff so please click on the Thank You link just below their post to thank a particular poster.
If you can't see the Thank You link below posts then either you are a new member and haven't acquired that privilege yet ... or you've used up all your Thank Yous for the day and need to wait till tomorrow.
The Thanks: line shows how many thanks a poster has left in total. The "Thanked X times in X posts" is exactly what it looks like. It tells you how many times a poster has been thanked. Someone with just one post could have acquired thanks from several different members and his record could show: Thanked 25 times in 1 post
Thank Yous left are publicly visible just below the post that users have left thanks for ...as you'll probably see at the bottom of this post in due course.
This is a bit trickier. There are two aspects to it - "reputation" and "reputation power". The former is the reputation points the user has been given, the latter is his/her ability to give reputation points i.e. how many points they give others when they rep somebody.
"Reputation" is the sum total of the reputation points you've received. Each time someone reps you your reputation score goes up by x points (depending on who it was who repped you and what their rep power is). "Reputation" is denoted by green boxes. You start with one green box and for every 100 reputation points you're given you gain one new green box in the initial stages. Getting new green boxes gets a bit more difficult once you've accumulated a few. You never know a user's exact reputation. You'll only know when he clears the next threshold to the getting another green box.
Rep power is the number of reputation points a user gives each time they rep someone. Your rep power increases based on how many posts you have, the age of your account etc. So, obviously, increasing your rep power is something you can do yourself, but your "reputation" increases only when someone reps you.
<added on April 30, 2012>Following a poll in the VIP Lounge in April 2012, I've decided to make a change to rep power. Previously, when you awarded someone reputation, the number of points you gave them was equal to your rep power. Now there's a cap at 15. The points you awarded now will be the same as your rep power till you reach 15. After that even though your publicly visible rep power increases, you'll still be awarding only 15 points when you rep someone. This is so "high rep power members" can feel more free to award rep and use the reputation system.</added>
[[OLD TEXT - Consider this now deleted: I've artificially frozen my own "reputation power" at 10 as it was too high relative to everyone else. I don't have a lot of "reputation" (just one green box at present) because I don't get repped often and when I do get repped it's by people who don't have a lot of rep power. Andy and hooperman are the only ones with two bars at present and that's because they've had a few reps from me - at 10 points a time. One or two others, like benitez17, are close behind them. I expect I will trail everybody else on this for a long while (which is fine. Don't rep me just to get my score up! )]]
When you click the star you're giving the poster reputation points. This does not diminish anything you've already got - neither your "reputation" nor "reputation power" is affected when you rep others.
Reputation is private. If you rep a user for his post other members wouldn't know about it, but that user's rep score would go up and he'd get notified.
How to use Thank Yous and Reputation Points:
No rules. You can thank people using either method. One's public, the other private. Asking for Thank Yous and Reps is obviously not in the spirit of the game.