There's a nice Irish pub in Arlington called J Gilligans, which is just off Abram Street. We recommend it for anyone from the UK who happens to find themselves in the Dallas/Fort Worth area1. To Americans, it's an authentic Irish pub. To a true Irishman from Ireland it's probably a good laugh at best. Great music though and good booze.
There are some other good pubs in D/FW too. If you're in Deep Ellum2, The Angry Dog in downtown Dallas is a good place to start your evening. They have a pool room, a stage with great music on the weekends (and some weekdays), and frozen Cape Cod.
Other great places to go in Deep Ellum include Club Dada, the Coppertank Brewing Company, Blue Cat Blues, and The Bone. There are many more places in Dallas to lubricate your cranium, and most of them aren't too bad.
However, the odds are that if you're from out of the country and you end up in Texas, you're not just looking for a drink: you'd probably like to see some cowboys. We hate to disappoint you, but most of the cowboys that you may have heard of are very dead. True cowboys don't really exist anymore and you certainly won't find any in Dallas. However, there is a bunch of wanna-be cowboys in the area, and the best place to find them is at Billy Bob's Texas, located near the Fort Worth Stockyards. You can get a drink there, and if you're really lucky, maybe you can take part in a good old-fashioned brawl!
We recommend learning how to fake a Texas accent, before you go, just in case.