Tomatoes, like peaches, are one of the many fruits and vegetables that will continue to ripen after they've been picked. They won't ripen quickly and they won't usually ripen perfectly—they won't ripen as well as peaches do, for example—but under-ripe tomatoes can definitely be improved upon. And best of all: it's beyond easy, so there's no reason not to give it a shot.
The Ripening Key: Managing Ethylene
As they ripen, tomatoes and other botanical fruits let off ethylene.
The more ethylene they're around, the more they'll ripen. So putting them in a situation where they're around more ethylene is the key to ripening them further.
Ripening Tomatoes Is Easy
Put all of your under-ripe tomatoes together in a breathable container—cardboard box, paper bag, plastic bag with holes cut in it—to help them ripen by capturing the ethylene they let off. Note that using plastic contains too much of what they let off; intact plastic bags and containers will trap in humidity along with ethylene and humidity will make the tomatoes rot, often before they even have a chance to further ripen.
In short: yes, you want to capture the ethylene, but you want a breathable container because you don't want them to get moldy in there!
Not Fast Enough for You?
If you want to speed things up, add a ripening banana. Bananas let off more ethylene than other fruits, so they help their brethren along the ripening path.
Other fruits work, too—avocados and apples are good choices to bump up the ethylene and move things along.
Keep an Eye on Them
Check the tomatoes regularly and remove them as they ripen. They are unlikely to ripen at exactly the same rate, so check each tomato by taking it out of the bag, feeling if it feels heavy for its size, looking at its color, and smelling to see if it smells like a ripe tomato.
Again, tomatoes ripened this way are unlikely to rival a tomato fully ripened on the vine, and tomatoes of poor quality can't really be helped much, but this method will help a quality tomato that was picked just a tad before its time.
How to Store Ripened Tomatoes
Once your tomatoes ripen, store them on the counter and use them as quickly as possible.
Tomatoes do not last long once they're ripe. Keeping them out at room temperature and in-view is the key to maintaining their quality and ensuring that you remember to eat them up sooner rather than later.