For travelling outside eastern Japan, there is no way to buy or reserve tickets from abroad.
(Unless you can read Japanese)
For eastern Japan, you can reserve train tickets on the JR EAST network via their website (found here).
NOTE: JR EAST only covers the following part of Japan:
Even if you can't reserve a train ticket in advance, you will find that unless you are travelling during one of the three busy periods, you will have no trouble finding a seat without reserving it in advance. Simply show up to the counter, buy a ticket (or show your JR pass) and the attendant will hand you a seat reservation.
The busy periods include:
- Golden Week: usually around 28 April to 7 May
- Obon: Usually 10 August to 20 August
- Shogatsu: 28 December to 4 January
As blackbird57 and Federico mentioned, if you plan on travelling by rail, a JR pass can be extremely worth it. Use websites like Hyperdia to get a rough estimate of the total cost of your train tickets and compare that to the cost of a JR pass, you could save lots of money.