Breaking Your Lease
If you are renting on a month-to-month basis, you need to provide 30-day notice to your landlord, preferably in writing. This means 30 days before your next rent payment (usually the first of the month).
ORC 5321.17(B) (http://codes.ohio.gov/orc/5321.17)
If you break a year-long lease, the general rule in Ohio is that you owe rent for the rest of the rental term.
There are a few exceptions:
You are starting active military service (50 App. U.S.C.A. § § 501 https://www.justice.gov/crt/50-usc-app-501-596)
The rental unit is unsafe or violates Ohio health and safety codes (constructive eviction), such as lack of heat or water (ORC 5321.07 http://codes.ohio.gov/orc/5321.07)
Your landlord harasses you or violates your privacy rights, such as sexual harassment, changing your locks, repeatedly entering without notice, etc. (ORC 5321.04 http://codes.ohio.gov/orc/5321.04)
Even if your situation does not fall within one of the three listed above you may be off the hook for rent for the rest of the lease. Under state law (Stern v. Taft, 361 N.E.2d 279 (Ct. App. 1976)), your landlord must make reasonable efforts to re-rent your unit. However, if the lease specifically reads that the landlord does not have a duty to mitigate damages, the landlord may still be able to recover these damages from you.
Generally, as long as you make an effort to find a replacement tenant who meets all of the application requirements imposed by your landlord, and communicate well in advance with your landlord about your intentions to break the lease, you have mitigated damages and you won't have to pay for the rest of the lease. If you are unsure, call an attorney or Southeastern Ohio Legal Services.