Also known as Cordon Tomatoes, Vine Tomatoes, or Tall Tomatoes, Indeterminate Tomato varieties have a single main shoot that grows indefinitely. In a perfect climate, this main shoot would grow up indefinitely. Normally, these are trained up a central support, like a cage or trellis, or stake. Some growers pinch of the side shoots that grow from the point between a branch and the main shoot. Others let these grow to provide more leaves (more leaves means more flavor) or more shade (more leaves mean more shade). The belief by the growers that choose to pinch them off is that they zap the plants efforts at producing bigger fruit (quality vs. quantity). Here in the hot northern California area, I go for the shade, since the sun burns the tomatoes here.
The tomatoes form on trusses growing from the main stem. Sideshoots that grow off of the main stem will also grow trusses, but removing these sideshoots will improve the vigor of the plant and the size and quality of the other tomatoes. In general, tomatoes from indeterminate plants are larger fruits, and some say that the added foliage makes for a tastier tomato than the determinate varieties. The fruit also ripen more gradually over the season, instead of all at once, like determinate fruits.
The shoots are leafy at the tips, instead of full of blossoms. The vines will continue to grow and produce until the frost kills them. In Redding, CA, they were still producing into November, when I pulled them down. I was just in time, too. A big frost occurred the next week.
These are best in areas where the season is longer, since they take a bit longer to initially produce, but they are definitely more productive and flavorful, due to the higher ratio of leaf to fruit.
Indeterminate varieties are great for growing under cover: greenhouses or polytunnels.
They are called indeterminate probably because there is no specific harvest time for these cultivars. The vines produce fruit all summer long, unlike the determinate varieties.
Not very many indeterminate varieties are grown in greenhouses, since they take up more room.
Dwarf Indeterminate Tomatoes also exist. Their vines stop at 3-5 feet long and are very decorative. With superior flavor and a long season, they are a good choice if you can find them.