Uses
There are a few reasons or uses that you would want to start at the center of a project or crochet piece. Examples of this include: granny squares, circles (either for something like a coaster or the beginning of a three dimensional shape such as an amigurumi).
Magic Ring or Magic Loop Method
To put it simply you wrap the yarn around your fingers a couple of times and then pull a loop of yarn through, chain up and then put your stitches into the center of the ring that you have created. Once all of your stitches are pu into the ring you can pull the tail to tighten it.
Pro: This method allows you to pull the center closed as tightly as you want it.
Con: This method does mean that you have to knot the tail from your magic ring before you weave in the end. This method can be difficult because if it is not done correctly the center will not pull closed and it is a little harder with this method to do it correctly.
Chain 4 Method
In simplest terms you chain 4 and slip stitch into your first ch. This creates a ring, you do your chain up after the slip stitch, do not cut off the yarn. Once you have done your desired number of chains you put all of your stitches into the middle of the ring created by your chain 4.
Pro: You can weave in the center end without having to tie an extra knot.
Con: This method does not allow you to pull the center ring closed as easily. If you will want this you would need to leave a longer tail before starting your chain 4 which will allow you to weave it through the base of all of your stitches into your chain 4 ring before pulling it tight, this also means that you would have to tie a knot to keep it this tight before weaving in the end. Also, the first few times that I tried this method I had a hard time figuring out what was the center of the slip stitched ring.
Chain Method
This method is somewhat similar to the method above, but not identical. To begin you are going to do the chain for the height of your stitch + 1. If you wanted to make the center fo your project out of single crochet stitches you would do your chain up of 1 for the height of your stitch +1 meaning you would ch 2 to begin. If you wanted to have the center of your project start out with double crochet stitches you would end up with a chain of 4, the 3 chains for the height of your first double crochet and then that additional chain.
Once this chain has been made you are going to put your desired number of stitches all into the first chain that you made (the chain closes to the tail/farthest from your crochet hook).
Pro: Easier to find where to put your stitches into as it is all just into a single chain space. Also, has a slightly tighter center than the chain 4 method.
Con: While still plenty sturdy, to me to use this method feels less strong than the chain 4 method or the magic ring method.
When to Use
How to know when you should use which method. The chain method is mainly used for bulky or super bulky weight yarns where the chain 4 method would be nearly impossible to execute due to the bulkiness of the yarn.
The magic ring method can be used on any type of yarn, bulky or not, but with chenille type/fuzzier yarns this method can strip off the fuzzy part making this method slightly more difficult to use. However, this problem can be solved by using a thinner type of yarn just for forming the magic ring and then doing your chain up and all of your center stitches with your desired yarn. however, if this sounds too complicated then you might want to avoid trying this when first starting to use bulky yarns.
I would not recommend using the chain 4 method for bulkier type yarns because seeing the stitches and the center of the chain 4 would get more difficult making it harder to execute.
In the end it really does come down to personal preference and comfortability with each of these types of starting center methods. It can come down to the type of yarn that you are using for your project. Some of this will be a bit of trial and error before you have one method that you favor over the others or for them to be executed without much difficulty.
