OK, Do we or don't we eat back the calories we worked off?

I eat back my calories and lose weight.

It's likely that you are inaccurate in either the amount of calories you are consuming (underestimating), or they amount you are burning (overestimating).

http://www.myfitnesspal.com/topics/show/1080242-a-guide-to-get-you-started-on-your-path-to-sexypants

Aside from including info about exercise calories, the above link is going to be a great source of info for you overall.

MFP is designed for its users to eat their calories back, but if you aren't weighing out your food and logging consistently, then you could easily be eating more than you think.

Also, it's important to remember weight loss isn't linear and you'll have days where you'll stall or even gain a little. And any change to your intake or exercise routine can cause additional fluctuations and it's best to give your body 3-4 weeks to adjust before making any other changes (if needed).

This.

I suggest eating back only a portion of your exercise calories if you use MFP, gym machine, or internet resources to obtain calorie burns, as they can be overestimated.

Simple terms "you eat to have a deficit" how big that deficit is and how you come about getting to that each week is upto you. 7000 calories = roughly 1kg of weight. Given calories burned and eaten are not an exact science you want to leave some room for margin of error. Pika also said it's not linear too so just cause you ate and exercised to a 1000 calorie deficit on a given day doesn't mean you've lost 1000 calories worth of weight it could take days to see the benefits of that day and what you are burning at the gym is what you've just eaten it works backward the last thing in is gonna be the first thing out.

Then you have fluids, muscle repair and mood added ontop that will affect weight whether you eat those calories or not

The first three weeks I was losing weight before I started exercising. Then, I started working out and would eat the deficit that I exercised. Now I seem to be gaining weight instead of losing. I'm only trying for one pound a week. What's going on?

Your muscles could also be retaining water due to the exercise. This will eventually even out (takes 1-3 weeks, depending on the person). Start by eating back 1/2 of the exercise calories for a couple of weeks and see where you are, then adjust if needed.

If it's cardio/HIIT - I eat back around 80%-90% - I use a HRM to track calories burned.
If it's Weightlifting - I don't eat back the calories.