See my posts related to Weaving here.

There are numerous types of weaving. Following are notes and photos of the methods I have some experience doing. 

Inkle - a warp-faced weave that is typically used for belts, straps, and similar. This is a modern style of simple loom, allowing a long finished length with a relatively easy warping method - medieval and colonial looms would have been box looms. String heddles are used to separate the sheds. More complex patterns are possible with supplemental warps. 

In the photos on the left, you can see the string heddles (white cotton) which pull down one half of the warp strands, creating a natural shed. To form the opposite shed, use your hand to pull down the non-heddled warp strands below the heddled strands (shown in the bottom photo). I used cotton yarn, in a solid purple and a variegated purple, to create a more complex look to the finished piece.

Cat optional.

Tablet or Card - also a warp-faced weave, also used for belts, trim, etc. But the resulting piece is thicker than inkle, because there are (typically) twice as many warp strands - one for each hole in the tablet. Sheds are changed by turning the cards, either individually or in groups. Patterns may be stranded in (threading the cards with different colors to achieve desired pattern, given a rotation scheme) or achieved entirely by rotating the cards, such as with double-faced weaving (each card is threaded with 2 strands of a background color and 2 strands of a foreground or pattern color; complex patterns such as lettering are possible with this technique).

You can see that the loom shown for card weaving is similar to the one shown above for inkle weaving. You can use the same loom for either, and there are several different styles available. My first tablet weaving projects I didn't have a loom, so closed the far ends of the warp in a cabinet door, and pushed against it with my feet. You might consider seeing if there are any fiber arts guilds or reenactment groups in your area, because someone would probably be willing to show you how to use their loom, and give you a feel for it, better than reading a book or watching a video would do.

Backstrap - My backstrap loom is fairly modern, though the technique remains the same as more traditional methods. You use your body to create tension on the warp, with a strap around your waist or back, which attaches to the close end of the warp, while the far end is secured to a table, post, or if it's a small project, your outstretched feet. Mine came with a nylon rigid heddle, and is great for making small pieces, such as headbands, garters, and similar. 

The finished product is comparable to what you get with inkle weaving - Merriam Webster defines inkle as "a colored linen tape or braid woven on a very narrow loom and used for trimming; also :  the thread used" - of course these days, inkles are made from all sorts of yarns.

This pair of inkles was made using my backstrap loom with ramie-cotton yarn, reclaimed from a thrift store sweater. They were made to be included in a gift basket, in my medieval recreation hobby (you can read more about that on my SCA page). I braided the ends, to give them a more finished look. With more recent pieces, I tend to twine the ends - it's a matter of personal preference.

Here you can see the two sheds formed by the rigid heddle. Rigid heddles are used in table looms quite a bit. Each different size of heddle gives a specific spacing, or sett. So if you want to weave different spacings with the same yarn, you'll need different size heddles.
In the larger floor looms, you'll often see wire heddles, grouped together in frames or harnesses. So an 8-harness loom would have 8 frames full of wire heddles.

Peg, Stick, or Straw - This creates a weft-faced weave, so instead of the long strands that create the length of your piece being prominent, the back & forth weft strands are most visible. This tends to create a softer piece, and can be used for belts, tapestries, rugs, and blankets, depending on the size of your loom. In the photo to the left, you can see the cotton warp peeking out from the strips of fluffy wool top. But once the piece is finished, the warp will nearly be invisible. With this type of weaving, you wind the weft back and forth between the pegs, and when you run out of room, you slide the weft down the length of the warp strands, which are knotted at the far end. 

Rigid Heddle
I bought a gently used rigid heddle loom a few years ago. I've woven a few scarves on it and need to play with it more. Earlier this year I purchased a stand from the man who made the loom. It's great for plain weave and at some point I'd like to add a second heddle for more pattern options. I belong to a great FB group for rigid heddle weaving - people are always sharing their projects, asking & answering questions. The biggest obstacle for me at the moment is that I don't have space for all my toys, so things get packed away and then aren't as easy to grab and work with them.

Four Harness
Shortly after I purchased the rigid heddle loom, a friend mentioned seeing a loom for sale at a local resale shop. When we visited, they had marked out the price and said "make an offer." I did, expecting it to be rejected as too low, but the loom had sat there for a few months without any other interest, so I got a Louet four harness loom for an excellent price. It sat unused for several months, because I didn't feel confident setting it up myself. Then I took a class at Traditional Arts Today, which was fantastic. I went home and warped my Louet, but after weaving a sample, noticed that my warp tension wasn't even across the sample. I need to go back to TAT and see if I can schedule a private session with one of the local weavers, so I can get back on track. While technically portable, this loom is heavy, so it sits on a table in the garage, with a blanket over it so the cats don't mess with it.