Why does my coriander plant look like this? Why are the leaves different? Is this what is called bolting? And why are they drooping?
Bolting is definitely not the problem here as that happens when a cilanto plant is mature. All of those plants in your picture are very immature. They look like sprouts, all fighting for space to grow.
I think that's why they are drooping, as the competition for space has made them grow too tall. None of them have developed beyond a slender stalk and cotyledons or a few tentative leaves, as Bamboo says in her or his helpful answer and comments.
This website shows in drawing 3 under "Growing Cilantro in a Pot" about how many seedlings you want in a container the size of yours. You want to space them more widely even at the start, and even then thin them as they grow.
You might want to try drastically thinning what you have there to maybe 8 to 10 plants spaced out a bit. (I suspect even those sprouts you thin out will taste good if you want to use them.) But those seedlings look awfully leggy so you might also want to consider starting over using only a sprinkling of seeds spaced an inch or so apart. That's a little hard to do with such tiny seeds, I know.
Cilantro is a short-lived plant and will bolt easily especially in hot weather. So you may want to stagger plantings even in your container so that you have a mix of older and younger plants growing at the same time.
We have grown cilantro in a variety of containers indoors and out and it is hard to know exactly how to maximize what you get during the growing season. Unlike a lot of plants you do need to be careful in how many seeds you plant and when, so as to spread out your harvesting.
But it's worth it to get fresh tasty cilantro all summer long and then let some plants bolt and flower to get some seeds to plant the next year or to use as coriander spice.
Whilst I can see some coriander foliage in the container, the other leaves are a different plant, not sure what. Trace the narrow, pointed leaves back to their point of origin - its probably something else that's growing in with the coriander. You may or may not be able to remove whatever it is by the root, but it looks as if the invader has largely taken over, so it may be next to impossible to extract it without destroying the coriander, unfortunately.
UPDATE: In response to your comment below. When did you sow these plants in that container, precisely, and did you sow lots of seeds? Because we might be looking at coriander cotyledons rather than true leaves. If that's the case, there are way too many seedlings in that container...
You sowed way too many seeds. That container will only successfully hold three, maybe four at a pus, mature coriander plants, so I reckon we are looking at cotyledons and there's insufficient room for the majority of the plants to develop properly. The drooping is likely due to overcrowding and insufficient water with so many seedlings trying to take it up all the time in too little soil. Next time, sow into a shallow tray and then transplant the seedlings into your container (and several more containers if you want lots of plants) when they've got two sets of true leaves.