The first screen after you tap ストーリー (Story) is this; chapter selection. You'll begin with only one at first, but after you complete the first level of every Robattle Mission, you'll unlock the next chapter.
第X章 means Chapter X. The text in the box is the chapter's name. See Chapter List for more information.
Each chapter has a series of シナリオ (Scenarios), or cutscenes or ロバトル (Robattle). Clear each robattle once and watch every cutscene to unlock the next chapter.
Every Robattle has three difficulties; ノーマル (Normal), ハード (Hard) and ベリーハード (Very Hard) in which the enemies get tougher, obviously.
Clear very mission on each difficulty to unlock a full-clear medal for that difficulty (bottom left of picture).