That is known as a versal or a decorative cap.

Or initial cap, or drop cap if it top-aligns with the rest of its line, or rubric if it's colored (I think).
Or birdletter if it's Armenian. :-)

hhp

Initial cap or drop cap don't convey the 'colorful or stylish background', since these may be plain.

A rubric is properly a piece of text, traditionally coloured red (hence the name), indicating appropriate movement or gesture during liturgy, e.g. 'The priest crosses himself with the paten'.

Love those Armenian birds.

It's also known as an illuminated initial and that can be further broken up into historic, foliate, and pen-flourished initials , etc.

Here's a good link: Illuminated Initial