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• Ai-gakari (相掛かり) or Double Wing Attack
  • Both players exchange Rooks' pawn at the beginning. There are various Senpo (戦法=Battle tactics) that belongs to Ai-gakari category:
    • Koshikae gin (腰掛銀=Sitting Silver) system
    • Kusari gama gin (くさり鎌銀=Chained Scythe Silver) System
    • Bogin (棒銀=Climbing Silver) System
    • Hineri Bisya (ひねり飛車=Twisting Rook) System
    • Various Yokofu Dori (横歩取り=Side Pawn Picker) Systems
• Kaku-Gawari (角換わり) or Bishop Exchange
  • Bishop will be exchanged early in the game.  Rook's pawn remain on the board. Again, there are several variants in this category.
    • Bogin (棒銀=Climbing Silver) This is not the same as Ai-gakari Bogin
    • Hayaguri gin (はや繰り銀=Rapid advancing Silver)
    • Koshikake gin (腰掛銀=Sitting Silver)
  • The above three systems possesses Paper/Scisors/Stone relationships to each other.
•  Yagura (矢倉)
  • The name came from it's castling formation. Neither Bishop or Rook's pawn will be exchanged in the beginning. There are so many Yagura systems. Explaining all would take up many pages. (I will add resource references later)  the link above shows very basic formation.)
  • I really like this opening. For what it's worth, here are more tactics.
• Furi Bisya (振り飛車) or Ranging Rook
  • Rook will be swung to your left side. Again there are so many variants of this "Ranging Rook" System.
    • Shiken Bisya(四間飛車=Fourth File Rook)
    • Naka Bisya(中飛車=Middle File Rook)
    • Sanken Bisha(三間飛車=3rd File Rook)
    • Mukai Bisya(向飛車=Opposing Rook)
    • Kyuusen Ishida Ryuu(急戦石田流=Rapid Ishida)
    • Jikyuusen Ishida Ryuu(持久戦石田流、or 石田流本組み=Orthodox Ishida)
  • If both players swing their rooks, the form  is called Ai Furi Bisya(相振り飛車=Ranging Rook vs. Ranging Rook). Ai Furi is becoming more popular these days. See here for demo games of Ai Furi

• More Resources on Shogi openings

Once you understand the fundamentals of openings, there are some more detailed openings shown in "Study Room" section of othis site.  

 

For further study, there is a Japanese site named "Chidori gin's Senpou Zukan" that lists more than 100 openings. 

 I was going to put together the navigation tips about the site. However, Takodori' san beat me to it and I don't need to do it any more!  He catalogued the site and posted them here and here.  Nice work!