Hi Sato2,
A lease is a contract (legal agreement) between you (tenant) and the landlord (owner) of the property. In addition to the rent amount, it covers included items (e.g. whether or not the landlord supplies appliances like a refrigerator or stove), property use and access rights, who is responsible for utilities, repairs and maintenance, etc.
The lease itself does not cost anything, however in some places you have to pay either a deposit or "last month's rent" along with the first month's rent at the time you sign the lease.
A 6-month lease means you agree to rent the property for at least 6 months - does not mean you have to pay for 6 months up front (can still pay monthly). But it does obligate you to pay the full amount. Say the rent is 1,000 per month, a 6 month lease means you ultimately agree to pay 6,000. If you leave early, you have to pay the difference between what you have already paid, and the 6-month amount. In some places, though, you only forfeit the deposit if you leave early.
A 6-month lease can be extended with a new lease, or if you stay past 6 months, you can go month-to-month with the same terms of the previous lease.
Many landlords want a 1-year lease, so if you're not sure about the place you want to live, a 6-month lease can be a good thing.