What does 'Hakuna Matata' mean?

Everybody remembers the popular motto from The Lion King, which was repeated in a song by a meerkat Timon and a warthog Pumbaa:

Hakuna Matata!
What a wonderful phrase
Hakuna Matata!
Ain't no passing craze
It means no worries for the rest of your days
It's our problem-free philosophy
Hakuna Matata!

By this song the two friends were trying to teach Simba to live without troubles and looking back at his past. The appearance in this movie made the phrase recognized worldwide. Actually, Hakuna Matata first appeared in a song almost a decade before the 1994 animated film. In 1982, the music of Kenyan hostel band Them Mushrooms released the Swahili song Jambo Bwana, which repeats the phrase Hakuna Matata in its refrain. The Lion King soundtrack was composed by Elton John and written by Tim Rice and was even nominated for Best Original Song at the 1995 Academy Awards. Later it appeared in the American Film Institute list of 100 best songs in movie history. Hakuna Matata took the 99th position.