Miter joints are nice looking. They are also quick and easy.
The trick to cutting nice miters is to get the angle exactly right. If you are making a (4-sided) box you need crisp 45 degree angles. Don't trust your miter gauge, make a few test cuts, then hold the corner together and check it with a square. If it isn't exactly 90 degrees your miters will be off.
Miter joints may look good, but they aren't particularly strong. They are fine on a small box that won't be used hard, but it's best to reinforce them. One of my favorite ways is to use a decorative spline.
All you need to do this is a jig to hold the box and/or lid at a 45 degree angle. I cut the kerfs for the splines on my table saw, but it can also be done using a router.
Once you've cut the slots you simply glue in a matching spline. Make sure the spline material is the same thickness as your kerf.
I have a drum sander that makes short work of this, but you can also cut the slot to fit the wood you have on hand - use a dado blade with shims as needed to get it just right.
No matter how you make them, the safest way is to cut a long strip the thickness that you need and then cut the splines from that.
If you prefer the splines to be hidden when looking at the box sides you can either use the same wood species for the splines, or you can make hidden splines. Do this while your saw is set at 45 degrees. First cut your miter joints as usual.
Then, cut the slot by laying the box side, inside face down, perpendicular to the blade. Use a miter gauge or sled with a stop so your cuts are all the same. You'll need to get the slot in the right place, so try a couple test cuts on scrap that's the same thickness as your work piece.
You want the kerf to be toward the inside of the miter where it is thickest. If you cut in the center you can weaken the miter joint by making the tip too thin. The goal here is to have the slot as deep as possible without cutting off too much material.
When cutting the splines for this type of miter it's important to pay attention to the grain direction. Unless you are doing something particularly funky, the grain will be perpendicular to the miter joint. Your spline needs to be also, otherwise it can split right along the miter and that rather defeats the purpose.
The piece of walnut on the left is correct for a hidden spline, while the cherry on the right should be used for a decorative spline instead.
Another fun trick is to make "faux" dovetails. This is the same principle as miter and decorative spline, except you cut a dovetail shaped slot and fit in a dovetail "key" rather than a spline. This is good and strong and will impress your friends and family (cutting actual dovetails by hand or machined dovetails will really wow them).
To make faux dovetails you will need a jig to hold the box at a 45 degree angle while you route a slot using a dovetail bit. Again, cut your miter joints first. Then go ahead and route the dovetail slot.
It's a good idea to start with a straight bit. Make a cut that is smaller than your finished dovetail, but that will hog out as much wood as possible. Do this in several passes as needed.
You should do this because when you switch to the dovetail bit you have to route at the final height. You can make several passes as you can with a straight bit. Rather than trying to route too much at once, start with your straight bit to get rid of the majority of the material.
Once you have routed your "dovetails" you need to make a matching key. To cut the key keep your dovetail bit at the same height and route on both sides of a long strip of wood (be careful with feed direction when doing this). Your individual keys will be short, but make them on a long piece for safety.
The strip that you use to make your "key" can be any length, but the width and thickness should be a bit more than your dovetail bit. The drawing is exaggerated so you can see what I mean.
Route a bit off each side sneaking up on the cut until the key fits the dovetail slot perfectly. Then rip the key off the strip and cut the keys to rough length. Glue in place and plane flush once dry.
These are a few of the fancier ways to strengthen a miter joint. You can also use plywood for the bottom of your box and glue it into place (if you use solid wood for a bottom, don't glue it or you will have problems with wood movement). The plywood will help keep the box together. This is especially useful on small boxes that just need a bit more strength.