The first pruning of the seedlings is done just a few weeks after transplanting them. It’s done so they wouldn’t grow like a bush, also by pruning, branches that remain will get more food and fruits on them will be bigger and healthier.
Pruning can be difficult to explain so I’ll just say it the way my parents explained it to me.

When you look at tomato seedling it should have normal branches like every other plant. But it has a sucker growing between stem and branch.

That little thing doesn’t do anything. It just takes food from plants so it should be removed. The process of removing it is simple just pluck it with your fingers. When they are small they are still very gentle so there will be no problems with removing them.

 If you do this on time it’s a simple job and it doesn’t take too much time.
Once the fruits set in it is time for second pruning. Second pruning is done to remove all the branches that will not give fruits and only take the food from the branches with fruits. 
And how can you know which ones will be fruitless?
When you look at the tomato stem you have to find the first tomato fruit. Everything under the tomato will be fruitless. Tomatoes never grow fruits under the ones that are already there.

You can remove all the branches under the fruit.  The next tomato flower/fruit will come above the ones that are growing now. By removing them the plant won’t lose the necessary food on unnecessary branches and food will go directly to fruits. Also, this is a good way to protect the plant from diseases. Most diseases come from lower branches. And of course, another plus is that the plant will not grow too many branches and turn your tomato bed into a forest.
There can always be branches that are already really big and look like they will have tomatoes. This is why I do this in July. If a branch that is coming out of the main branch(the one with tomatoes) by the end of July doesn’t have flowers or at least buds, then you can remove it. It won’t have time to mature and make fruits.

After you clean your tomato it will look like this. You can see how everything under the tomatoes is clean and above are normal(untouched) branches.
After I pick up these tomatoes and new ones come I will clean the next level.
It’s like cleaning the stairway…step by step.

I guess by now everybody knows this but I will say this again. Pruning is only for indeterminate tomatoes. DON’T prune determinate tomatoes.Ok, now that I said that I’ll say this you can prune them. Just not all of them.
There are some determinate tomatoes that grow like a bush(fruits come on new stems) and there are some of the tomatoes that grow similar to indeterminate tomatoes.
They grow big stems with just a few branches that get bigger and grow tomatoes on levels. Once they have reached their height they will stop.
For example, this is Don Juan tomato. A determinate plant that can be pruned. 

Campbell is also one of these tomatoes. The one you can’t prune is for example Oregon spring. It grows stems with tomatoes like a bush. If you prune him you will end up only with fruits that are already there. It will not grow new ones.