Tomato suckers really do suck. In my garden, they are a lot of work to deal with. Probably the most work of all other things tomato.
But what are tomato suckers? Tomato suckers are side shoots that grow between the stem and leaves. Their hope is to become another main stem...if you let them. If you let them grow, they eventually become another stem full of more flowers, leaves, and of course, more suckers. They'll give you more fruit too.
I don't want them because I'm trying to grow as many different tomato plants as possible in a small space. If you don’t want them, pinch or snip them off while they’re still small.
In my garden, I prune off most of the suckers. Even though I have a big backyard, and a big garden, it’s never big enough to grow everything that I want to grow. And I have 130 tomato varieties this year! So I grow my tomato plants really close together, sometimes only a foot or less between them. Then I make them to grow up--instead of out. By keeping my plants to one stem, I can easily twist that stem around twine, tie it to a stake, or weave it up through a cage.
Not everyone pinches out suckers. Not pinching suckers can increase your yield, and save you time and work. If you go to a market garden, you might see tomatoes with all sorts of stems (suckers that were allowed to grow) growing all over the ground (instead of up a stake). And that's OK if you have the space!