Here is a description of the most frequently used notation for Hive, which is also the one used on this website.
Each piece used during a game has a unique name.
This name is made of one lowercase letter for the color (w for white or b for black) followed by one uppercase letter for the bug type (A for Ant, B for Beetle, G for Grasshopper, L for Ladybug, M for Mosquito, P for Pillbug, Q for Queen Bee or S for Spider).
In case the bug appears multiple times, the piece name also contains one digit indicating in which order that piece came into play.
For instance, wA1 is the first white Ant that has been added to the hive while bB2 is the second black Beetle. Since there is only one copy of it, the white Pillbug is simply named wP and not wP1.
A move is indicated by the name of the piece that did move (even indirectly when using a Pillbug) followed by its destination position.
Since Hive has no board, it is tricky to uniquely identify destinations. Instead, we use one of the adjacent pieces as reference and one symbol (either –, / or \) to indicate where the destination is relative to this reference piece:
For a piece (like a Beetle) ending on top of another piece, the latter is used as reference, without any additional symbol.
Here are some examples:
Note that the very first move is necessarily different because there is no reference piece available to use. In this case, the destination is either a dot (.) or it is omitted.
In addition to these "normal" moves, there are also "special" moves to represent player interaction that does not change any pieces. The special moves are pass, resign, offer-draw, accept-draw and decline-draw (with obvious meanings).
This way, a game can always be represented by a list of (normal or special) moves.
The best way to get familiar with this notation is to view a game (try for instance this one) and follow the move column on the right as pieces get moved around.