DETERMINATE - This means the tomato plants stop growth at a certain point. As a general rule, they have short main stems, form fairly compact bushes and take up less space in the garden. There are short growing determinate varieties that are adapted to container growing, such as Tumbler Hybrid Tomatoes or the new AAS Winner Terenzo Hybrid Tomato. There are 'in-between' determinate types such as Siberian or Bush Celebrity that can be grown unstaked or in pots. And there are taller determinate varieties such as Celebrity Hybrid Tomato and Viva Italia Hybrid Tomato that are normally grown in large wire cages, tied to short stakes or allowed to sprawl. Determinates tend to bear their flowers and fruit all at the same time, a good feature if you like to harvest fruits over a short period for canning or freezing. While many will start to decline after they fruit, the newer hybrids may continue to bear all season or will produce a second crop, especially if the vines are cut back after the first crop is picked. Technically, the plants produce two leaf stems, then a cluster of flowers/fruits, then two more leaf stems and a cluster, etc.
INDETERMINATE - These tomato plants just keep getting taller and taller, continuing to set more and more fruits as they grow. They are normally tied to taller stakes or large wire cages unless your garden is large enough to allow them to ramble. To obtain larger fruits, pruning can be done to force the plant's energy from plant growth to the fruit. This may reduce yield, but the tomatoes will be larger. Indeterminate tomato plants will keep fruiting until they are killed by frost or some kind of disease. Technically, they have three leaf stems, then a cluster of flowers/fruits, then three more leaf stems, a cluster, etc.
There are advantages to both types of tomatoes, so many gardeners grow some of each. In a nutshell, determinate types conserve garden space and produce large crops all at once, while indeterminate types spread out the harvest season, usually producing higher yields overall, but requiring regular training with stakes, tying and cages to keep them growing in the right direction.