Apply for a Certificate of Eligibility.
A veteran who doesn't have a certificate can obtain one easily by completing VA Form 26-1880, Request for a Certificate of Eligibility for VA Home Loan Benefitsand submitting it to one of the Eligibility Centers with copies of your most recent discharge or separation papers covering active military duty since September 16, 1940, which show active duty dates and type of discharge.

Decide on a home the buyer wants to buy and sign a purchase agreement

Order an appraisal from VA. (Usually this is done by the lender.)
Most VA regional offices offer a "speed-up" telephone appraisal system. Call the local VA office for details.

Apply to a mortgage lender for the loan.
While the appraisal is being done, the lender (mortgage company, savings and loan, bank, etc.) can be gathering credit and income information. If the lender is authorized by VA to do automatic processing, upon receipt of the VA or LAPP appraised value determination, the loan can be approved and closed without waiting for VA's review of the credit application. For loans that must first be approved by VA, the lender will send the application to the local VA office, which will notify the lender of its decision.

Close the loan and the buyer moves in.