1.
be ...ing (action in progress)
With verbs that describe an activity lasting over time (読む, 走る, 待つ), 〜ている says the action is going on now: 本を読んでいる (I am reading a book).
2.
be in the state resulting from ...
With verbs describing an instantaneous change (死ぬ, 結婚する, 開く), 〜ている says the change has happened and the resulting state continues: 窓が開いている (the window is open), not "is opening."
3.
habitually; repeatedly
With a time expression or a plural object, 〜ている describes a repeated or habitual action: 毎朝走っている (I run every morning).
Formed from the te-form of a verb plus いる. Which meaning appears depends on the verb, not on 〜ている itself: activity verbs give the progressive reading, change-of-state verbs give the resulting-state reading.
Forms
- 書いている: is writing
- 書いています: is writing (polite)
- 書いていない: is not writing
- 書いていた: was writing
- 書いてる: contracted form, very common in speech
Watch out
- 死んでいる means "is dead," not "is dying"; 知っている means "know," not "am knowing." For verbs of instantaneous change, translate with an English state, not a progressive.