It starts by find a property that you like. You can do this in a variety of ways including using a realtor, looking through MLS listings on your own and working with the seller's realtor, looking through "For sale by owner" listings, etc."
As part of this, you can also work with mortgage lenders to do a pre-qualification. This process is basically where they, sometimes, will pull a credit report, get some info from you on your income and your estimated down payment, and tell you what they believe the price range of home you can qualify for will be. This can vary based on the mortgage program that you will end up using, as programs like FHA loans or first-time home-buyer loans might allow you to provide a smaller down payment. Keep in mind, though, that you will need to put some money down. I believe the lowest is 3 or 3.5% on an FHA loan. Typically anything under 20% will require some sort of mortgage insurance, whether it be FHA or private just depends on the loan program.
Once you have found a home you like you will submit an offer. There are many ways to go about this and the negotiations you and the seller get into are going to depend on you both and your associated willingness to negotiate. The market right now is relatively tight, so you cannot always count on the ability to submit or receive a counter offer or two.
Once you have an accepted offer you will work with the mortgage company to submit the necessary documents for mortgage approval, an appraisal will most likely be done, title work gets ordered, you will need to establish homeowners (also called Hazard) insurance, flood insurance if you are in a flood zone, etc. You may also need to work with a realtor and/or attorney to do things like get a survey or deal with some state-specific items like smoke certs, certification of occupancy, etc.
After you have a mortgage approval, you will typically do a final walkthrough of the property on or near the day of your closing and then you sign the documents and hand over the check or wired funds to cover the remaining closing costs/down payment.
Then it's time to move in.
That can sound complicated, but finding a good realtor, mortgage company, and/or attorney can help you walk through it pretty easily. There are things that will vary, but that's the general process for buying a house with a mortgage and down payment.