Tex-Mex is that hybrid of Mexican and American cooking that birthed the fajita, and nachos were born directly across the border. Burritos are not terribly common here unless you frequent the chains, but you'll find a wide range of tacos and enchiladas and tamales, as well as some surprises. Texas has its borderlands and some of its cuisine is so authentically Mexican that the fare may be a little unfamiliar.
Isn't that the fun part of border towns? If you want Mexican food so authentic that it's in . . . Mexico, it's just a border-hop away. Take Del Rio. On one side of the border, you can have all the Tex-Mex staples: fajitas, chalupas, and enchalidas. On the other side of the border, you can have those as well, but if you'd like a taste of goat, Portuguese-style chickens, or roast quail, Acuna, Mexico the place for you.
Or take El Paso, where the restaurants do serve up a lot of Tex-Mex, and not just the staples. Many restaurants serve unique dishes and some change their menu constantly. And if you're in the mood for spicy red snapper or brochette, you walk across the bridge to Cuidad Juarez.
Laredo is also most definitely a border town and its Mexican sister, Nuevo Laredo, is another place to try yourself some excellent red snapper. However, Laredo's cuisine isn't quite like all the other cities that specialize in Tex-Mex. It's a town where the menus offer local takes on old themes. For example, a Laredo staple is a breakfast taco called a mariachi, and oh is it spicy. Another Laredo innovation is the pancho, nachos covered with cheese, beef, and jalapenos. Or you may run across the machito, a goat-tripe sausage. If that last doesn't sound so attractive, there's plenty else on the menu, and plenty of restaurants with regular Tex-Mex offerings.