Twist is what holds the fibers in your yarn together. If you are a spinner or if you've ever watched anyone spin, the basic idea behind spinning is that you take some fiber and add twist and you get yarn. It's that simple. And that complicated. Because the amount of twist added to make the yarn is a variable that can greatly affect the yarn and what you make with it.
Softness vs. durability
One of the big topics for yarn is the softness vs. durability spectrum. Unfortunately, there is no perfect yarn that is both super soft and super durable. There are yarns that are more toward the middle of the spectrum (I think BFL is a good example of that) that are a decent compromise, and there are yarns that are more toward one end or the other (e.g., Merino toward the softness end vs. something like Leicester Longwool toward the durability end). Ultimately, you need to decide what quality is more important to you for a specific project and pick a yarn that fits that.
But the fiber content isn't the only thing that impacts a yarn on the softness vs. durability spectrum. Number of plies is one factor, and I wrote about singles yarns a few weeks back. But twist is another big factor, and the basic idea behind it is that the less twist a yarn has, the softer it will be; the more twist a yarn has, the more durable it will be.
So you can take something like a super soft Merino and twist it super tight and get a much more durable yarn. But when you do that, you're moving it along that spectrum toward durability and away from softness. It's quite possible to add so much twist to Merino that it feels more like rope than what you'd expect from Merino.
And I can tell you that from experience. I'm a spinner, and a while back I spun up some gorgeous Merino. But the yarn I made had so much twist in it, that I feel like I "ruined" the Merino. It was not at all soft. But the colors were gorgeous, so I decided to make a cowl. I don't know what I was thinking because there was no way I would want to put that rope-like Merino around my neck. So I thought about it and realized that the stitch I was using had a lot of texture to it and combining that with the high twist of the yarn, it was a very thick, structured fabric. So I continued with the cowl-like tube and then added a bottom to it to create a bowl.
So if you're wanting a project that is more durable, choosing a yarn with a higher twist is a good option. For example, a high-twist yarn for socks would be a good choice. Looser twist might be good for a cowl so the yarn plays more to the softness you'd want around your neck. A high-twist yarn would be great for a sweater as long as you're not wearing it next to your skin.
Twist and gauge
Twist can also affect gauge, so you'll want to take that into consideration when picking a yarn for a project. Have you ever substituted a yarn and tried to match the designer's gauge but find you're not even close no matter how many needle sizes you go up or down? It's not the only factor in that issue, but twist could be part of what is causing you not to match gauge. See if you can research (Ravelry is perfect for this) the yarn used by the designer to determine if it's a high-twist, low-twist, or mid-range twist yarn. Then choose your yarn to match that twist level as much as possible. (I do have upcoming article(s) planned about substituting yarn.)
How much difference can this make? Well, I have a fingering weight yarn (Merino) that has a tight twist vs. one with a lower twist. Now they aren't completely equal as one has 3 plies and the other has 4 plies, so that will make a difference also. But the gauge difference between the two is rather startling.
The high-twist yarn (orange) gives me a gauge of 34 stitches and 52 rows over 4 inches. The mid-range twist yarn (grey) gives me a gauge of 28 stitches and 40 rows over 4 inches. (I used the same needle size for both swatches.) That makes a huge difference in your project!
Twist is energy. The more twist in a yarn, the more energy it has. You might have experienced a yarn that likes to almost kink up while you are using it; if so, that's a high-twist yarn. It just can't help itself and has to "move"! A low-twist yarn is rather "lazy" and can seem almost limp.
When working stitch patterns where you want stitches to stay where you place them, a high-twist yarn might not be as cooperative on you. Look at these two cable swatches: the high-twist yarn on the left makes the cable not quite as "neat" as the mid-range twist yarn on the right. Were I making a project with lots of cables, I would avoid that high-twist yarn as I prefer my cables to look like the one on the right. (Low-twist yarn is also not great for cables as it'll lack the structure needed for that type of stitch.)
There are tools you can use to measure the twist in a yarn (most of these tools are made for spinners, but they can be used to measure commercially spun yarn as well), but you don't need to be technical about it. Just take out some yarns from your stash and start comparing some to see how much twist one has vs another.
Something else you can look at is the yardage in the skein of yarn. All things being equal (fiber, thickness of yarn, and number of plies), the yarn that has more twist is going to have fewer yards than the yarn with less twist. Why? More twist means the yarn is more tightly compacted so there is more fiber within a yard of yarn, so that yard is heavier. For example, most likely a 4-ply Merino fingering weight yarn with only 400 yards per 100 grams is going to have a tighter twist than a 4-ply Merino fingering weight yarn with 435 yards per 100 grams.
There's so much more that can go into the topic of twist, but this should give you a foundation to keep exploring and experimenting. If you're a spinner, or even if you aren't, a back issue of PLY Magazine covers the topic of Twist and is quite fascinating.