Kendo is an artform that requires a partner. However there is a lot of time when you can't train with a partner but you can still help your kendo improve by training by yourself. Suburi is the main way of doing this.
Suburi is shinai-swinging practice and can be done in the air or with a target to strike. Most kendo dojo start training with some form of suburi. Some dojo just do a few quick repetitions and others spend a long time on it. Some sensei find suburi very useful, others feel that it emphasises the wrong aspects of the cut, especially if done without an actual target.
Although it is great to have a target to strike (like a hitting dummy or uchikomidai), I am going to focus here on kuukan datotsu, or striking empty space. This is by far the most portable kind of training you can do by yourself, you can do it anywhere and it has a variety of purposes. You don't even need a shinai or bokken.
If you live in an old house with high ceilings (and no low-hanging lights!) then you can easily do suburi inside, standing up with a shinai. Just be careful of vases and people... If you live in a newer house with low ceilings then you can still practice suburi a couple of ways.
1) With a shinai: sit in seiza and do sets of sho-men uchi and sayu-men uchi. Increase the difficulty by sitting in sonkyo and performing suburi that way. Alternate katate (one-hand) and morote (both hands) suburi.
2) Without a shinai: you can use a well-stoppered champagne bottle filled with sand to practice katate suburi standing up and using normal footwork, even in a house with low ceilings. This is also an excellent form of strength training as a bottle full of sand is very heavy! (Caution: this is not something you should try until you've been doing regular suburi for a number of weeks).
Outdoors offers more options with your swinging weapon: extra-long suburito can be used if you have one, and the ground surface does not matter too much. Hayasuburi (jumping men) is also possible now. Normally a fairly even surface is best, however it can be useful to do suburi on uneven or sloping ground from time to time.
Even though you are just striking the air, you should always focus on your imaginary opponent (gasso teki). Try not to fall into the habit of looking at the ground when you practice, but look up, as if at someone who is about the same height as you. Performing suburi in front of a full-length mirror is a good reminder of this.
How much is enough suburi?
There is an old adage that if you cannot train with a proper teacher, then 1000 suburi by yourself every day will still see you become a good swordsman. A thousand suburi every day is certainly a very good target to aim for, but build up to it. Don't set yourself the task of starting at 1000. You will just give up sooner.
If you are new to kendo, or have not done many suburi for a while, start off with a small number, but do them EVERY DAY. If you do, you will naturally build up the number of repetitions as you feel your arms, shoulders and wrists getting stronger. Start with fifty or less (per day) in the first week, a hundred in the second week, two hundred in the second week, and so on.
How much is enough also depends on your age. I would say for primary school aged kids, 50/day is plenty. For junior high school kids 100-300 is enough. And for senior high school kids and adults 300-500 is probably plenty to maintain good kendo strength.
You don't have to increase the number of suburi to get more benefit. There's a big difference between doing 1000 suburi in lots of 100, 200, 400 or all in one go. You will find you maintain your interest if there is some variety. So alternate between one hand and both hands, joge, sho, sayu, hayasuburi (jumping men).
It's easiest if you think in sets of 100. After a while it becomes quite easy to keep count, but at first you might find you keep losing track. Keep at it and when in doubt, add an extra 10 or 20 for good measure.
The importance of the left hand
If you only have time for 100 suburi per day (and that only takes about 2 mins, so everyone can do that much if not more), I would strongly advise that those 100 be left-hand only suburi. As most people in their first 3 years of kendo have a fundamental flaw in their swing based on using too much right hand, left hand only suburi is the only way to remedy this situation. Never, in your first 5-10 years of kendo, practice right-hand only suburi (people specialising sei-nito are an exception here) only left-handed katate suburi. The reason for this will become clear to you if you do the practice.
Also, practice your katate suburi straight up and down: sho-men uchi is the best suburi for the left hand. Keep a close eye not only on how straight your cutting action is, but how straight your backswing is too. Again, a mirror is excellent for this. If you don't have a good mirror but you do have a digital camera, recording yourself in movie mode from the front is almost as good (a mirror is better because it's instant feeback).
Why do suburi?
Suburi is good for building and maintaining strength in your kendo muscles, especially in your left hand and arm. For children a little bit of strength training is a good thing but don't overdo it. Teenagers who are not naturally strong, particularly girls, will really find their kendo improves from a technical point of view. They might also find kendo becomes more fun because they achieve more winning cuts by having a sharper cutting action.
As you get more experienced, strength becomes secondary. Suburi becomes more like what practicing scales is to a musician: they keep you in touch with your instrument. Even if you can't get to training, you will notice a big difference in your kendo if you do suburi in between trainings, as compared with if you do nothing.
Keep checking with your sensei that your technique is good. Doing regular suburi at home is only useful if it is correct suburi.
How much is too much suburi?
There is such a thing as overtraining, but most people are so far from that they need not worry. Kendoka preparing for the All Japan Championships regularly do 2000-3000 suburi every day, on top of their regular dojo training schedule. If you can maintain a reasonable number, say between 300 and 800, every day, then there is little chance you will pick up an injury, especially if you vary the suburi, vary the weight of the shinai or bokken, vary the number of suburi with you do without a break, and so on.
Of course if you do notice pain in your shoulders, arms or wrists, and it seems to get worse the more you do suburi, then stop. See if the pain also stops. Listen to your body and if you think there's a problem, go see a doctor or physio whom you trust.
Suburi is a simple, repetitive excercise. It's not very interesting by itself. But then it's not the reason most people do kendo. What is fun is seeing how your own kendo improves through regular suburi practice.