Yes—and here are some of the basic steps involved. Keep in mind that an experienced lawyer's help will be crucial to successfully preparing for, filing, and winning a housing discrimination lawsuit against your landlord (as well as filing a complaint with an administrative agency, such as HUD, or negotiating a settlement outside of court).
Federal, state, and local housing laws all prohibit various kinds of discrimination against tenants. The most common kinds of discrimination—race and disability—are prohibited by the federal Fair Housing Acts (see Housing Discrimination Prohibited by Federal Laws for details on "protected categories" under federal law). Most state laws also make the same forms of discrimination illegal, and in some cases go even further—for example, by prohibiting housing discrimination based on sexual orientation or gender identity (groups that are not covered by federal law). Local anti-discrimination laws may also come into play. (See Housing Discrimination Laws Prohibited by State and Local Laws for details.) The book Every Tenant's Legal Guide, by Janet Portman and Marcia Stewart (Nolo), covers different types of illegal discrimination and tenant strategies for fighting back.
Tenants (current and prospective) have several ways to fight unlawful discrimination—including negotiating a settlement with a landlord (perhaps with a mediator's help) and filing a complaint with a government fair housing agency that may result in a conciliation agreement between landlord and tenant. For an overview of your options, see the Nolo article How to Fight Housing Discrimination.
The more serious harm you suffered, the more likely you are to win your lawsuit. Landlords who practice intentional discrimination, such as making flat-out statements that they don't rent to African Americans, are far more vulnerable to a fair housing charge than someone who unintentionally discriminates by letting only two people rent a two-bedroom apartment because they don't understand fair housing laws on occupancy limits.
If you want a landlord to rent you a particular apartment, you may find that filing a complaint with a fair housing agency is the best way to go; one result may be a conciliation agreement in which you agree to drop your complaint in exchange for the landlord's written agreement to rent you the apartment. If you're seeking damages for emotional distress caused by a landlord's discrimination, or punitive damages for especially blatant and intentional discrimination, a lawsuit may well be your best bet.
You may file a lawsuit in either federal or state court. If you have a strong case and can get a quick court order from a judge, your experience will be mercifully short. Or if your facts are compelling and your lawyer good, you may be able to settle for a sizable amount of money without going to trial. But not every case proceeds so easily. Landlords (and their insurance companies) may dig in their heels, resulting in months, or even years, before a case is settled.
HUD's State Information section is a great place to begin. Click on your state, then on "Get Rental Help," for a list of local organizations that provide a variety of housing services.
You'll clearly need the help of a lawyer experienced handling discrimination cases if you plan to go directly to court to force the landlord to rent you the premises and pay you money damages. A lawyer can also help if you file a complaint with a fair housing agency. You may file a lawsuit in federal or state court even if you have filed a complaint with a fair housing agency (deciding whether to do both, and timing everything, is one of the reasons you'll need to hire a good lawyer). And a lawyer will be invaluable in negotiating a settlement agreement with a landlord, whether or not you end up filing a lawsuit.
Legal aid attorneys, hired by government-financed law clinics, may be available if you have very low income. But, for many people, hiring a private attorney will be your only option. If you have a strong case, an attorney make take it "on contingency" (you pay nothing up front but agree to give the attorney a percentage, such as 30% of whatever you win or agree to in a settlement). Before getting started, read the Nolo articles on hiring and working with an attorney, including how to find free or low-cost legal help and other tips about lawyer fees.
And be sure to check out Nolo's Lawyer Directory to find landlord-tenant lawyers in your state with experience representing tenants in housing discrimination lawsuits.