I have a rigid-heddle loom and have liked weaving on it, but I'm really interested in weave structures I see in Handwoven, such as overshot, doubleweave, and summer and winter. I'm thinking this means I should get a loom that can do those weaves. Which loom should I get?
I'm afraid the only right answer is "it depends." It depends on what you want to weave and what your resources are in terms of time, money, and space. Your options, more or less, are a table loom, a jack floor loom, a counterbalance floor loom, or a countermarch floor loom.
Table looms come with anywhere from two to sixteen shafts. They have the advantage of being able to raise any combination of shafts without the restrictions of floor looms (the number of treadles limits the number of different sheds you can make). They have the disadvantage of requiring more time for weaving (moving the different levers with your hands instead of stepping on treadles with your feet). They therefore don't allow a weaving rhythm, the way floor looms do. They are less expensive than floor looms (usually), they take up less space, and they are portable. You can't weave really wide or really long fabrics on them (no rugs or tablecloths).
Floor looms come in three basic types: jack, counterbalance, and countermarch. (I'm talking manual floor looms here, not computer-controlled looms.)
With jack looms, when you step on a treadle you raise a combination of shafts. The advantage to this is that you only tie treadles to the shafts that go up. The disadvantage is that the shafts can be heavy and the treadling can therefore be heavy, depending on the width and make of loom. Jack looms are available with as many as sixteen shafts.
With counterbalance looms, shafts are connected to each other in such a way that when one goes down, the other goes up. The shed is made by moving threads both up and down, therefore providing a more even tension on the up and down threads than you get with a jack loom, and also greater tension. Counterbalance looms are therefore ideal for weaving rugs. Some counterbalance looms do not allow an unbalanced shed (one shaft up, three down). Almost all counterbalance looms are limited to four shafts.
Countermarch looms can have as many as sixteen shafts, though most are limited to ten. Every treadle is tied to move every shaft, either up or down. The disadvantage is therefore that you have to make many ties (80 ties if you are using eight shafts and ten treadles). Shafts are moved both up and down, allowing even and maximum tension. The treadling is also light as for a counterbalance loom. Shed adjustment, however, can be a bit tricky, especially if you use eight or more shafts.
It's a really good idea to go to a school or shop where you can try out a number of looms so see what you like. Join a weavers' guild, where you can get lots of advice and visit looms at weavers' houses. Guilds love to encourage new weavers. Happy loom shopping!