There's a nice video floating around the Web of one of these dingalings who sell knives on Home Shopping Network. He whacks this cheap imported sword on a table -- presumably to show how strong it is -- at which point the sword breaks in half and the tip flies into his chest. I don't think he got a punctured lung or anything, but it sho nuff ruined his day, I guarantee you.

Technique

Hasuji is probably the most important ingredient in tameshigiri.   Hasuji is the term used to describe correct blade angle. The cross-sectional centerline of the sword should enter the target on precisely the same angle as the swing. Just having your blade enter the target a degree or two out of alignment with the angle of your swing can ruin your cut.  Hasuji is improved by paying close attention to proper grip or tenouchi.

Tachisuji. This is the angle of the entire blade (as opposed to hasuji, the cross-sectional angle).  There are various schools of thought on this score. Some people think that a cut should "draw" against the target, whereas others believe that developing maximum speed by "chopping" is preferable. Each has advantages. A chop works well against firmer targets, whereas a draw cut works particularly well against resilient, soft targets like leather or dry fibers -- for instance, paper or textiles.

Make sure you practice different kinds of cuts. Cut down on an angle to the left and right. Cut up on an angle from both sides. Cut parallel to the ground. Before you ever cut a target, you should have mastered these cuts technically by practicing suburi (making practice cuts in the air). As you begin to master the basic cuts, you can do things like drawing and cutting in one stroke, multiple cuts, multiple targets, one-handed cutting, etc.  (Not to belabor the safety issue, but advanced techniques increase the chances of injury, so make sure you know what you're doing before trying them.)

Types of targets

Tatami. Traditionally test cutting in Japan (when not performed on dead bodies) was practiced on bundles of wet rice straw and on bamboo. Now most schools that do cutting use rolled up tatami omote (thin reed mats) and bamboo. Tatami is an excellent test of technique. Poor technique will reveal itself in the form of incomplete cuts, scooped cuts, bent or squashed tatami rolls, and (occasionally) bent or otherwise damaged swords.

The problem with cutting tatami mats is that they cost an arm and a leg. To really get good at something, you need to practice it every day. Unless your last name is Rockefeller, you probably can't afford to chop up tatami every day.

Bamboo. Bamboo generally doesn't cost anything, but most of us don't have a huge stand of it in our back yard to harvest. Which means that if you want to practice regularly with it, you probably have to turn into a bamboo bandit. Which has its disadvantages. ("What am I doing sneaking around on someone else's property at midnight with a machete in my hand? Well, gosh, Officer, it's kind of a long story...")

So what else can you cut up? And to what end?

Wood. Wood is a highly underrated target. Especially if you have a table saw or a band saw, you can buy ten bucks worth of lumber from Home Depot, cut the board up into strips and cut them. You'll learn a lot. You can get quite a few cuts out of one six foot long piece of 1" by 1". In some respects, it's actually a more serious test of your blade than tatami. And it's a lot cheaper. It's also a good test of your sword. If you cut wood and the edge gets all dinged up, you know you've got a crappy sword. Hang it on the wall and do not cut with it again. Ever. Tatami will not teach you this lesson until the sword breaks. Which is really not how you want to find out. I don't recommend trying to cut 2X4s, by the way. Stick to smaller pieces.  One inch by one inch is a good size.  Beyond that point you're just putting more stress on the blade than it's really designed for.   Also, I recommend using softer woods like aspen and poplar.  Avoid knotty woods like pine.  Pine knots are extraordinarily hard -- hard enough to damage sword edges.  Unless you're cutting fairly thin sections, stay away from hardwoods like oak, hickory and maple.

Cardboard shipping tubes.  It takes a real man to cut a heavy weight four inch cardboard shipping tube in half. Smaller tubes work well, too, and are a lot less hard on your sword. One thing you have to watch out for with hard objects is that if you cut through too vigorously, you may end up bouncing your sword off the floor. Not a good idea.

Plastic jugs.  Water-filled plastic drink jugs are an interesting target. Everybody in America throws away eight or ten of them a week. They aren't that hard to cut, but they do test the sharpness of your sword. If you can't cut a two liter Pepsi bottle full of water, then you know your sword needs sharpening. Like all cutting tests, they also help you develop good aim. And they're kind of fun. Problem is, you had better be extremely meticulous in cleaning your sword, or you'll have rust problems. Hint: the less water in the jug, the harder the cut.

Newsprint.  Cutting through a single piece of newsprint formed into a cylinder is a very demanding test of sharpness and blade geometry. It's actually easier to do with a very slim knife than a sword.

Meat.   Kinda weird.  Kinda creepy.  Highly recommend you don't try this in front of the neighbors.  But darned instructive.   Swords, after all, are intended to cut meat.   Cutting meat will give you a whole new level of understanding about why you don't want to goof around with a sword.

Stuff not to use.  Don't test your sword on rocks, glass or metal objects. I read a post on a web site by a guy who said he used his sword to cut steel cable. The guy was either a liar or an idiot. Trying to cut anything that is as hard as – or harder than – steel is just asking for trouble. First off, it'll destroy your blade edge. Second, it may break your sword. In which case you have created a missile weapon with the mass of a thirty millimeter depleted uranium cannon shell. Those puppies will blow up a T-72 tank. So don't kid yourself into thinking that the last ten inches of a sword couldn't put a hole clean through your head.

Equipment

First, make sure you have a sword that's suitable for tameshigiri. I can't stress this enough!  Never use anything made from stainless steel.  It's too brittle. Never use an iaito (a blade that's only intended for iaido practice). Stay away from cheap imports.  There are some excellent Chinese-made swords, but the majority of imported swords are wall-hangers that should never be used to cut with. (Unfortunately there are unscrupulous people selling wall-hangers as real swords. If you're not confident of your ability to tell the difference, find somebody who can determine if you have a real sword or a wall hanger.)

It helps to have a good tameshigiri stand. The standard tameshigiri stand is made of a 4" by 4" post, just under waist high, with legs that radiate out from the stand.

A few final notes

First, always clean and oil your sword after cutting. If you don't, you'll eventually end up with a rusty steel bar.

Second, be smart. Don't rush. Don't cut things that are beyond your skills. Don't freak out your neighbors by acting like some ninja weirdo. Etc., etc.,etc. Swords are extremely dangerous weapons and using them in foolish ways can cause you to: hurt yourself, hurt your friends, scare your neighbors, get sued, end up in jail, get shot by Officer Friendly.  These are all suboptimal outcomes.   Be smart, be smart, be smart.

Finally -- and most importantly! -- I strongly urge anybody with an interest in swords to find qualified instruction in a sword art. There's a lot more to using a sword than whacking up beach mats in your back yard.