Like with any country there is a lot of minor variation on how to go about funerals and this applies to weddings as well.
The example that NewbieRunner provided is, I think quite accurate particularly in a corporate context. His example seems to represent quite large funerals. I just wanted to add a few things to be aware of for smaller funerals where it isn't the company member but his or her family member for whom the funeral is for.
It is getting increasingly common in Japan to have smaller funerals and to these it is usually family members only, quite frequently even girlfriends/boyfriends aren't invited, although there is no invitation as such and the information usually filters down to the relevant people. If you become aware of the death and you have a close enough relationship then you would contact them to express your condolences. If the date, time and place are mentioned to you then I guess that counts as an invitation of sorts. You can always ask and if it is for families only they will be clear about this. Bear in mind that it is not unusual to have meals and a room prepared for the attendants so going completely unannounced can cause logistical problems. If you do go unannounced then be prepared to leave after offering incense. Also very important, like Newbie Runner mentioned - take money - I would say at least 20,000 but ideally 50,000 (avoid 40,000 as number 4 can be pronounced shi which means death in Japanese and hence there are no number 4 rooms in hospitals). The money has to be placed in a special envelope - to somebody new they don't look that different from wedding ones so be careful and which way the bow is arranged is very important too. Most will come with instructions at the back. You will need to write your name on the envelope (again follow instructions). This is done so that the family can keep track of who came and what gifts they can send to you afterwards, thanking you for attending the funeral.
Just as a final note, I think the example of 410,000 yen that the company contributed in NewbieRunner's example is a symbolic gesture to avoid the whole number 4 problem - I guess they felt that 300,000 wasn't enough and 500,000 was too much or the people that couldn't attend contributed and ended up being 400,000 and someone added another 10,000 in order to avoid the 4 - ! Finally, you always increase the amount to avoid the 4 and never down.