It means 'No worries for the rest of your days'. Everyone now knows the meaning to the Swahili phrase 'hakuna matata' thanks to the Lion King. The song, written by Elton John and Tim Rice, and originally recorded by Nathan Lane and Ernie Sabella, was one of the main plot points and one of the catchier tunes in the film. Timon and Pumbaa explains to Simba how they live their life.
While I love this song, it always sort of bothered me that this song gained so much popularity because it is teaching Simba a lesson that he later finds out is holding him back from achieving his goal. Not really the lesson to teach kids.
This version of Hakuna Matata can be found on the 1995 release RHYTHM OF THE PRIDE LANDS, a follow-up soundtrack to the popular LION KING soundtrack. The album features LEBO M, a South African composer who did a lot of work on the Lion King movies and Broadway musicals. Joining him on this track is JIMMY CLIFF who you probably as the guy who sings I Can See Clearly Now. You'll notice that this version not only expands on the origin of Pumbaa but it also goes into Timon's past.