All attacks while mounted are normally made as actions or reactions using the stand and attack action/reaction.
The mounted character is allowed to react to his/her mount's movement bringing a target within his/her strike range (and is allowed to react part way through his/her mount's movement) or to a target's movement action bring the target into his/her strike range.
The hexes into which a character may attack with a melee weapon are modified as a result of him/her being mounted. A mounted character sits in between two hexes on the combat board and is partially obstructed by his/her mount (so can strike more easily on his/her weapon side than on his/her offside). These factors affect which hexes he/she can attack into, as shown in the diagram below: