Summary: the board of directors is a group of people selected by the stockholders that get together as necessary to make high level decisions of the company. The board is different than the officers of the company that manage day to day affairs. 

What is the board of directors?

The board of directors is a group of people that oversee and protect the interests of the company. The size of the board differs based on the needs and size of the company. The board plays a powerful role in the company. 

Who selects the board of directors?

The stockholders of the company. Financing agreements will often provide guidance regarding the voting procedures and requirements that dictate which classes of stock elect which directors. 

What does the board of directors do?

As mentioned, the board oversees the company. It doesn't do the run of the mill daily tasks. It doesn't actively run the corporation. That is left up to the officers of the company (e.g. CEO.) It does make big decisions of the company. 

The board has certain legal duties to the company such as the duty of loyalty and the duty of care. 

When does the board of directors meet?

The board meets at the least once a year (this will be detailed in the company's bylaws.) The board usually meets periodically--at least once a quarter or even monthly. It all depends on the needs and requirements of the company. 

Does my company need a board of directors?

Yes. Every company is required to have a board.

If your company is in its infancy and it's just you and a co-founder and there's no funding in place, don't freak out thinking you need to find people and create some big fancy board. 

You can have one director of the board if you want (and this person can be the same person as the founder.)  As the company gets bigger and raises money, the company will increase the size of the board.

How big is the board of directors?

The board size is flexible.

If the company has done one round of financing, 3 directors is a good number. The sweet spot for most VC financed companies is 5. But it's not uncommon to see between 5-9 directors as the company grows.

Who is on the board of directors?

The typical set up is half are designated by the investors, half by the common stock holders, and the odd one an outsider that is mutually agreed to by the investors and common stock holders. If the board is sizable (i.e. 7+) then more outsiders makes sense.

What kinds of decisions does the board of directors make?

As mentioned, the board makes high level decisions of the company as distributions to shareholders, raising money, adopting stock option plans, etc.

What does or doesn't need board approval depends on the company and the company's bylaws. If it's a high level decision, it probably needs to be approved by the board. 

How active is the board of directors? 

This is extremely variable. I've seen some directors that are extremely hands off and leave it to the officers and personnel of the company to do everything. I've seen some directors that want to know every fucking detail under the sun. 

A balance between active and hands-off is best in my experience. 

How does the board of directors make decisions?

By getting together and voting on matters. But this isn't always necessary. A written consent can be given in lieu of a formal meeting. 

So let's say that some decision needs board approval. It's a simple decision, but for whatever reason the board needs to sign off on it. Everyone on the board agrees with the action. There's no reason to get everyone together to do this. So instead of doing a formal meeting, all board members sign adopt the resolution by signing a written consent. A corporate lawyer/startup lawyers assists with drafting these resolutions.

You can imagine other situations where signing off on a document is feasible. For example, if you're the only board member of your company. 

What is the difference between a board of directors, advisory board, and board observer? 

Board of Directors 

See above

Advisory Board 

Basically just a bunch of advisors. That's it really. Nothing special. You don't need one.

Board Observer 

A person who has the right to attend board meetings but has no right to vote on matters. Board observers should generally be discouraged as they can strongly influence votes. You don't need them. 

Advisory board and board observers basically comes down to two things:

Concessions with board politics and making people whose opinions you value feel special. 

There's nothing inherently wrong with making people feel valued. And it's good to get the opinion of people who have good insight and who you can trust. However, having a designated advisory board can make things unnecessarily complicated and costly with getting people together, consulting, stock options, etc. There may be times when a new board member needs to be brought on (e.g. new investor demands it) and it causes issues with prior investors/prior board members, so as a result a board observer designation is created. 

In general, avoid an advisory board and board directors. 

Should my company pay board members?

No. Giving board members stock options is fine however particularly if they are outsiders (i.e. they have not already invested in the company).