Open and half-open files
A file is one of the eight vertical columns on the board, labeled a through h. A file is open when neither side has a pawn on it anywhere -- nothing blocks a rook or queen from sliding all the way from one end to the other. A file is half-open when only one side has a pawn there: it's still clear for that side's opponent to use, but a lone enemy pawn stands somewhere on it.
Rooks and queens are long-range pieces that get stuck behind pawns, so a file with no obstruction is exactly the kind of terrain they want. A rook parked on an open file can swing all the way to the back rank, pressure whatever's sitting there, or later invade the 7th/2nd rank from it.
A half-open file is still useful, just slightly less so: your rook can use the whole file freely, but the single enemy pawn sitting on it becomes a natural target, since it usually has no pawn of its own left to trade off or defend it with.
Because seizing the best open file is a long-term positional goal rather than a one-move trick, a common middlegame plan is to maneuver a rook (sometimes both rooks, 'doubling') onto the open file before the opponent can contest it or close it up.
Worked examples
Open and half-open files side by side
The d-file has pawns of both colors, so it's neither open nor half-open. The f-file has only a white pawn -- half-open. Every other file has no pawns at all -- fully open.
Fully open files (a, c, d, f, h)
Pawns only sit on the b-, e-, and g-files (both colors). Every other file -- a, c, d, f, h -- has no pawns at all, so it's fully open for either side's rooks.
A single half-open file
Black's only pawn sits on e7 with no white pawn to match it -- the e-file is half-open for white. Every other file has no pawns of either color, so it's open too.
Test yourself
Click any square on the one fully open file.