Conventional mortgages have historically been the most common mortgages in the industry. They are backed by Fannie Mae and Freddie Mac who are government sponsored agencies, and tend to have the same qualification requirements across the board.
Conventional mortgages have various down payment options. When putting a smaller amount down, you will have an added requirement called PMI (Private Mortgage Insurance). The amount you will be required to pay for PMI is determined by several factors: Loan-To-Value, credit score, down payment, and other additional factors. Also, the ways in which the PMI can be paid vary; up-front, monthly, and lender paid. Conventional mortgages do not require PMI when you put down a large enough down payment.
Conventional mortgages allow for fixed rates, and adjustable rates. With a fixed rate mortgage, the amount owed for principle and interest every month will not change throughout the life of the mortgage. With an adjustable rate mortgage your rate will remain fixed for a predetermined number of years, after which point the rate can adjust to an indexed rate with predetermined margins for adjusting.
Also, conventional mortgages do not typically have a pre-payment feature which gives you the flexibility to pay additional towards principle every month without being penalized, as well as paying the mortgage off in full without being penalized.