There is already a question about making mortise and tenon joints using hand tools. It would be nice to also catalog the various ways of making mortise and tenon joints using power tools. Wherever a jig is required for safety or quality of fit, please also include a photo or link to the type of jig mentioned.
Copying my answer from the 'by hand' question.
The mortise as the other have mentioned can be done with a hammer and chisel, and drilling consecutive holes to the correct depth will help speed the process up.
If you have a drill press that will make the job easier and you can be very accurate on how deep you drill all the holes. If you are using power drill, you might want a collar for the drill bit for depth. Here's an article for step by step directions with pictures for the mortise.
If you have a little extra money to spend you can buy and actual mortising machine, it drills square holes. It has a drill bit with 4 chisels surrounding it.
Cutting the tenon. There are many different ways to do this.
Of course there is the handsaw, routers etc. You can also use a table saw.
The table saw can be used several different ways, the easiest would be if you have a Dado blade so you can run the pieces flat across the saw table.
You can do it without the dado as well, just make a bunch of cuts and then clean them up with a chisel. At the very least you can cut the shoulders of the tenon on the table saw, giving you an nice square cut to work with.
There are also tenoning jigs for the table saw which you cut the face of the tenon with one pass, so a full tenon would be 4 passes. Kind of like this:
Long mortises and tenons, like the following, are probably best created with a hand router and a table saw.
For the mortise, you will need
A jig or brace to ensure a straight path for your router.
A straight router bit with a flat blade on the top will provide the necessary cut. However, a spiral-fluted bit will make the cut easier.
Obviously, make sure the diameter is less than or equal to the width of your mortise. If it's less then you'll have to adjust your jig or set up a second one corresponding to each edge.
- The router will leave rounded edges to the mortise. In order to square them up, you'll have to use a hand tool like a chisel.
Image from acornhouseworkshop.com
As mentioned in other answers, a matching long tenon can be made with a table saw, a dado blade, and a tenoning jig.
To make the mortise, you can get a dedicated mortising machine:
You can also get a mortising attachment for your drill press:
Image courtesy of Lowes
Note: Some people feel that these attachments require too much force be applied to the quill and handle of your drill press. I've used my attachment on a bench-top drill press a few times and don't recall any excessive amounts of pressure. If you're making a lot of mortises, you'll probably want to invest in a dedicated mortising machine anyway.
Either machine uses a mortising chisel to cut a square hole:
The tenons can be cut on a table saw using a tenoning jig. Here is a commercially available one:
And here is a shop built version:
Another alternative is to make and use a horizontal mortising machine. It's much easier than you might think.
I use mine to make loose tenon mortise and tenon joints. It's a joy to use. There were 18 mortise and tenon joints per chair in my dining room table set project. That's 108 joints. That's far too many conventional mortise and tenon joints to be practical. So I built this:
The vertical face closest to you holds the sliding fence with the router.
Here is the sliding fence that carries the router:
This is based on a youTube video at https://www.youtube.com/watch?v=knk4vWXI5ZQ0
Notes: 1) The video uses sliding dove tails. I couldn't get them to slide smoothly so I used drawer slides. Because drawer slides are very loose, I added four machine screws in each table and adjusted them to ride on steel bars as shown above (just visible on the far left under the handle). This holds everything level and tight.
2) The router is a cheap Chinese router from a well known tool chain store.
3) I added a foot switch from the same source. It's very convenient.
4) My first version flexed too much. I built this version from 3/4" MDF and added a third bolt which runs in the same slot as the primary height adjustment bolt.
5) How to not lose all 5 fingers on one hand at once: DON'T FORGET TO DISCONNECT THE FOOT SWITCH before adjusting the height of the bit.
"Shop Improvements" from Taunton Press page 163 shows a simpler but less capable alternative.
A comment on one of the answers mentioned Mattias Wandel's pantorouter. Somebody else said they should add it as an answer, so here it is.
Wandel's pantograph is a more complex take on a typical router pantograph, only horizontal and with plunge so you can do multiple passes and variable depth.
First, Wandel's website (which has lots of fun woodworking projects, btw):
Wandel actually started with this simpler (but still pretty nifty) slot mortiser:
What I like about both this mortiser and the pantograph, compared to the many other slot mortiser designs I've seen, is that they make it much easier to make larger mortises. I mess around with viking-style furniture based on the Oseberg bed ( http://www.currentmiddleages.org/tents/osbergbed.PDF ) which requires a lot of mortise & tenon joints that are much, much larger than most of the stuff I can find on M&T.
However, Wandel's slot mortiser only does simple rectangular holes. The pantorouter can do complex shapes, just like any pantograph, and you can use it to do both mortises and tenons.
Oh, note also that Wandel's pantograph has a 2-1 size ratio, i.e. you cut the pattern twice as big as the actual result. This is nice because it shrinks any errors in your pattern.
Here's one of the videos I found a more effective demonstration of the pantorouter. I added a time parameter to jump to 4:12 where he cuts a tenon and then a matching mortise. Watch it from the start if you want more detail:
And here's a video where he explains how the pantorouter design works:
He has lots of other videos that go into building it, etc, like this one:
One of Wandel's fans started producing (with his permission) pantorouters made from metal. Last I looked they were $1200 and had to be shipped from Japan, but that was a few years ago. A quick google just turned up this video Wandel did, explaining how to assemble it:
I've always wanted to build one of these, but then again this doesn't look like a trivial project at my skill level. I mean, the reason I want to build one is that I'm not that good at woodworking :-). And I'm more interested in the result (building stuff I can use) than in the process. Don't get me wrong, I'd like to be highly skilled at this stuff, but clearly spending the actual time and energy required to get there has not yet made it to the top of my priority list :-).