What's the difference between a character, a code point, a glyph and a grapheme?

Character is an overloaded term that can mean many things.

A code point is the atomic unit of information. Text is a sequence of code points. Each code point is a number which is given meaning by the Unicode standard.

A code unit is the unit of storage of a part of an encoded code point. In UTF-8 this means 8-bits, in UTF-16 this means 16-bits. A single code unit may represent a full code point, or part of a code point. For example, the snowman glyph (☃) is a single code point but 3 UTF-8 code units, and 1 UTF-16 code unit.

A grapheme is a sequence of one or more code points that are displayed as a single, graphical unit that a reader recognizes as a single element of the writing system. For example, both a and ä are graphemes, but they may consist of multiple code points (e.g. ä may be two code points, one for the base character a followed by one for the diaeresis; but there's also an alternative, legacy, single code point representing this grapheme). Some code points are never part of any grapheme (e.g. the zero-width non-joiner, or directional overrides).

A glyph is an image, usually stored in a font (which is a collection of glyphs), used to represent graphemes or parts thereof. Fonts may compose multiple glyphs into a single representation, for example, if the above ä is a single code point, a font may choose to render that as two separate, spatially overlaid glyphs. For OTF, the font's GSUB and GPOS tables contain substitution and positioning information to make this work. A font may contain multiple alternative glyphs for the same grapheme, too.