Form
It is often shortened to 〜てる
Ongoing action
Shows an ongoing state (progressive or perfect)
働いている I am working.
current state
it can also be used to express a state that results from an action. 瞬間動詞 (instantaneous verb) cannot be ongoing, so the uses never conflict.
結婚します I will get married (not 'am married')
結婚しています I am married (in a state of after-marrying)
窓のガラスが壊れている The glass of the window is broken (it broke, and it still is broken.)
For some verbs however, such as 着る, they can conflict, or be the same.
セータを着ています I am putting on a sweater ○ I am wearing a sweater (after having put one on) ○
病院に行ってる **I went to (and still am at) **the hospital.
You did something, and it has a lasting effect
ている should be used if an action in the past greatly affects the present
昔、日本に住んでいる I lived in japan in the past (and it still has an effect on me).
〜ていた
When conjugating it to the Past tense, it can have two meanings: You were doing something (following from Ongoing action, or you have already done something (from current state). Which one it is depends on signal words like もう.
昨日の11時妹は昼ご飯を食べていた Yesterday at 11 AM, my younger sister was eating lunch.
昨日の11時妹は昼ご飯をもう食べていた Yesterday at 11 AM, my younger sister had already eaten lunch.
〜ておる - Humble
In very formal documents often used is the 謙譲語 (Humble language) form of this expression
本製品は、フォントワークス株式会社のフォントを使用しております。 本製品は、フォントワークス株式会社のフォントを使用しています。 This product uses fonts from the Fontworks company.
japanese.stackexchange.com - ている vs ておる
Visualising Japanese Grammar 25 - resultant て patterns
YouTube: Miku Japanese - Common mistake in Japanese and how to use ている and ています