Skills refer to the various abilities of himes, be it atk up / def up/ double attack rate up buffs for your allies and/or atk down/ def down etc debuffs for your enemies.

Frame of skill is like their "type", mainly for ease of keeping track the stacking compatibility between them. Typically, there are the A frame and B frame. A stacks with B, but A does not stack with another A. Rarely, you will find some other types, like C frame, eg hime cthulhu, hime thanatos, or Eidolon frame, eg Yamata-no-orochi's active ability def down. That's why those himes are valuable because their C frame skills stack with the common A frames and B frames, granting you additive bonus.

To identify the frames of skills, look for them in the DMM wiki. If you don't read Japanese, Google translate does a passable job in indicating frame types, but the skill descriptions can get wonky. Also, frame types do not correlate with the border color of the skill icons in game.