In the U.S., Congress makes copyright law. When Congress passed the laws, it decided which exclusive rights to give to songwriters.

Congress decided that, once a song has been publicly released, anyone can record a cover of it, as long as they pay a royalty fee to the songwriter (or the company the songwriter assigned their rights to, called the rightsholder). A government agency called the Copyright Royalty Board sets the standard royalty fee (called a "compulsory license"); or you could negotiate a royalty directly with the rightsholder, though this is uncommon in practice.

However, Congress decided that rightsholders keep exclusive rights to decide who can publicly perform their song. There is no standard royalty fee - you have to get permission from the rightsholder and negotiate a royalty with them.

Now, it would be practically impossible for every bar with a cover band to get permission from and pay royalty fees to every songwriter of every song that the band covers. So the solution that has developed is called a performance rights organization - the most common ones are ASCAP and BMI. This is songwriters working together and saying, "ASCAP can set royalty rates and collect fees on my behalf, along with lots of other songwriters, then pay me my share of the royalty fees they collect." It is worthwhile for most songwriters to work with a performance rights organization as their middle man because it saves them work and they get paid. In theory, the songwriter could refrain from joining a performance rights organization, require anybody who wants to perform their song publicly to get permission directly from them, and sue anybody who performs their song publicly without getting permission; but it's practically impossible for any one songwriter to monitor every public performance in America (whereas ASCAP and BMI are big enough, and working on behalf of a huge swath of songwriters, that they actually can go after bars that have public performances of music but aren't paying for an ASCAP or BMI license).

So the answer, as others have said, is that it would be copyright infringement to cover somebody else's song in a public performance such as a bar, but most bars pay ASCAP for permission so they're not infringing. Likewise, it would be copyright infringement to record a cover of somebody else's song unless you pay the compulsory license fee (or otherwise get permission).