After opening Wikipedia one too many times just for copying out a character, I figured it would be easier to just put it all in one place. To that end I made Gliffic, a handy Unicode reference webapp. It searches the Unicode character database by name and codepoint, and displays the results for easy copying.

Gliffic allows users to search codepoints by partial name or hexadecimal value. Results display the glyph, hexadecimal codepoint, and canonical name. Emoji are rendered in color.
Three completely reasonable searches with Gliffic