I like to grow open pollinated, heirloom indeterminate tomatoes. Because indeterminate tomatoes grow and produce all season I like to thin out the suckers from my tomatoes. I know this is a huge question with tomato growers with many different opinions on the subject. For us it helps to keep our very large tomato vines inside their cages, well supported and makes picking a little easier. It is basically just trimming the branches of the tomato. You may get fewer tomatoes but the ones you get seem to be healthier and are easier to care for and pick. I think it also helps with improved air circulation within the bush and fewer disease problems.
I tried to label on this photo the main stem, the main leaf and the sucker. Notice where the main leaf attaches to the main stem that there is a shoot right in that junction. They usually are very vertical. I simply take and snap that off when they are very small. This helps to thin the plant and the plant can concentrate on tomatoes instead of growing all those leaves.
Notice on this picture above where I have snapped that sucker off right there in the junction. This doesn't harm the plant at all.
Do you sucker your tomato plants?
Blessings from The Holler
The Canned Quilter