The original title of this play was Magee Monolgue as Beckett originally wrote the play for an Irish actor named Patrick Magee. The final title, Krapp's Last Tape, like the original, suggests that this is going to be a play centered around one man.
The name, Krapp, is a also a homophone of "crap," a crude slang word common in Britain, which as a noun means excrement, and as an adjective is used to describe something that is worthless, or at best of a poor quality. Krapp's name, therefore, perhaps suggests that the character may be somewhat unfortunate, or that perhaps he either feels or is perceived as worthless.
The key word in the title, "last," implies that this will be Krapp's final tape. He makes one every year, as a sort of annual, audio diary entry, but in the play he is rather elderly, celebrating his sixty-ninth birthday. During the play there are also suggestions that his time is perhaps running out. He sings, for example, these lyrics from a song: "Now the day is over, / Night is drawing nigh-igh, / Shadows - of the evening / Steal across the sky." The implication is that Krapp is old and close to death, and that the tape he records this year will be his final, or last tape.
The word "last" can also be read to mean the most recent. This meaning also ties in neatly with the play, as Krapp is indeed settling down to listen to previous tapes. In this sense then the word "last" might suggest that Krapp's most recent tape is the most important, perhaps most revelatory or somehow most crucial tape.