Margin is basically a loan. You have a brokerage account with $50,000 in it. However, you want to buy $75,000 of a hot stock you know is just about to skyrocket. Margin lets you purchase that extra $25,000 worth.
Brokers feel fairly confident about making margin loans, because the stocks purchased (or whatever other asset is getting bought) becomes collateral for the loan. If things go south, the broker issues a margin call...basically forcing you to sell the stock and repay the loan.
However, while brokers are confident, they aren't crazy. You can't just wander into a brokerage with a top hat and monocle and demand a margin loan on the force of personality. You have to fulfill some requirements.
Enter: minimum margin. It's the lowest amount you have to deposit in an account to be eligible for a margin loan.
The minimum margin for an account at your favorite broker is $10,000. You deposit $9,500 in there to open an account. No margin for you. But bump that up to $10,100...suddenly, you've got access to those margin loans.
Actually, that broker has a pretty high limit. The NYSE and the NASD both require a margin minimum of $2,000 in cash or securities.