Promoting a pawn to the eighth rank simply refers to moving a pawn across the chess board without it being captured in the process. This is generally a late-game strategy, as there will be fewer threats on the board. You do have to watch out for your opponent's pieces and protect the pawn in some cases, due to the danger your pawn will present to your opponent, should you promote it.
Chess is too complicated of a game to write an exact walkthrough for this. This achievement really shouldn't be an issue, however. If you play for long enough, this achievement is essentially a given. I obtained this after 3~ hours of gameplay, though I'm sure I could have obtained it in less than an hour, had I been playing against the easy A.I. and actively working toward it.