が居(い)るTo be, There is, To remain (in a state)

がいる is a common construction that is simply a combination of the が particle, and the verb いる. This expression means 'to exist', or 'there is', and is used in relation to living/animate things.

Because the things being described in these examples are considered to be alive (or operated by something that is alive), いる is the verb that will be used. ある is used for non-living things.

The polite form of いる conjugates as います. Due to the る being removed, we can determine that いる is a る-Verb (Ichidan verb).

The kanji form of 居(い)る is quite common, and should be learned as early as possible to save any confusion while reading. There is no nuance difference between this expression being used with or without kanji, and is up to the preference of the writer.

Caution

The が in がいる is often omitted in casual speech, or writing that behaves like casual speech (manga, instant messaging, etc).