The oldest and most common type of flying club is called a shareholder club. Members purchase a share in the club with their initiation fee and actually own a percentage of what the club jointly owns. The club takes care of maintenance and other fixed costs out of monthly dues. Members pay the variable costs (fuel, for example) with an hourly rental fee.
A leaseback flying club works a little differently. It has a minimal initiation fee because the planes are actually owned by investors who purchased them for lease back to the club. Monthly dues are usually about the same as a shareholder club. The hourly rental fee can be slightly higher, but it is still less than an FBO's rental fee.
The major difference between a shareholder and leaseback flying club is what you get for your initiation fee. Shareholders actually own a share of all the aircraft. That means your membership costs more—and has some resale value. It's an asset.
A variation on the leaseback flying club is the cooperative flight center. Depending on how the co-op is set up, the primary advantage is that you can buy an aircraft with a relatively low initial investment (under $10,000) as the main asset of a "flight center." Other pilots in the area then can become members and rent the aircraft through you at reduced costs. Because it is a business there are some tax advantages to this structure in addition to cost savings. For further information, contact Flight Management Services or check aviation magazines for ads on other co-op ownership programs.