I like Tom's answer but for someone not involved in the industry you might want a little more detail.
Loans where the amount lent is less than 80% of the value of the underlying property are considered conventional loans. The lenders relies on the ability to sell the property at less than a 20% loss if the borrower fails to pay the loan off.
Lenders can get insurance or guarantees on loans they make. When a loan is insured or guaranteed, it can readily be sold in the secondary market, which is where FHLMC (Freddie Mac) and FNMA (Fannie Mae) operate. If the loan is insured with private mortgage insurance, it is a conventional loan.
If FHA guarantees the loan or VA issues a guarantee or USDA, then these loans are considered government-backed loans, not conventional bank loans. FHA charges a mortgage insurance premium (MIP) both upfront and each month until the loan drops below 80% LTV.
The secondary market for government-backed loans is quite active these days, because the losses banks suffered making conventional loans with PMI were staggering. Only recently have banks really started making more conventional loans.
Also, Fannie and Freddie have quite an inventory of foreclosed properties, as well as HUD (the parent of FHA) and VA.
The requirements for a conventional loan are similar to government-backed loans. You need good credit (620+), cash to put down on the house, and sufficient income to pay back the loan. Conventional lenders use similar ratios to determine if you have sufficient income, and similar down-payment requirements (but higher than FHA).
FHA wants 3-1/2% cash from a buyer-borrower. Conventional loans usually require 5% to 10%, although some of their foreclosed properties can be bought with 3% down. Rates are similar to FHA, perhaps a shade higher. The mortgage insurance (MI) wasn't available for a while because the PMI companies basically lost a lot of money. Now the MI rates are a little higher and graded by risk (credit score).
From your perspective there really is no difference in the loan paperwork and income requirements etc. But, you may not like the higher down payment and slightly higher interest rates. For borrowers with low loan-to-value (LTV) ratios, there really isn't a difference.
If you're a veteran, the VA offers loan guarantees on loans with no down payment. Texas offers special rates on loans to disabled veterans or normal rates on all other veterans but with lower fees -- the TexVet program is not VA, which is Federal.
You may not be a veteran, but out in Fate you can get a Department of Agriculture loan (USDA), which is very similar to the VA program, including zero down payment. USDA loans are restricted to rural areas.
FHA has no geographic limitations and has the smallest down payment for non-veterans and those in non-rural areas. Conventional loans are generally available everywhere, too.
Does that help?