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The answer to this question depends on where in the world you live. In the United States, then once a song has been recorded and released, then anyone can record and release their version of the song, without permission. A statutory compulsory royalty is payable.
Outside of the United States then generally you will need the permission of the copyright owner to record the song, and of course a royalty will be payable.
There is an interesting article here, in which a musician describes the process of becoming a skilled acrobat and jumping through a amazing sequence of hoops to release his small-scale recording of song covers.
In general however, your best course of action is to go to the people in charge, which will be the agency or agencies responsible for the administration of music royalties in your country.
In the USA go here...
And as the article linked above suggests, if the Harry Fox Agency says no, you can always go direct to the publisher because you have a legal right to record previously released material.
Also useful for finding out who publishes a particular song in the USA is the Music Publishers' Association.
We would love to hear from anyone who has an interesting experience of licensing music. Also if anyone has fallen foul of copyright law and has been made to pay the price, we would love to hear about that too.
P.S. One of the comments below correctly states that in Canada, CMRRA is the agency responsible for administration of mechanical rights, and they should of course be your first port of call in that country. I am pleased to add it to my article. However, the linked article describes the potential difficulties of finding out who owns the rights to a particular song. I know from my own experience that the databases of these agencies are not entirely complete. So check with CMRRA first, but SOCAN might also be a useful contact.