Fonts are made up of a set of character shapes, or 'glyphs', together with size and spacing information known as 'metrics'. Glyphs consist of vector shapes, which work in a similar way to the lines, curves and ellipses you might draw in Illustrator - they're defined by geometric instructions, not a grid of pixels, so they can be scaled to any size.
Dating from an earlier age of computing, PostScript Type 1 fonts were limited to 256 characters. That's plenty for standard Roman upper and lower case, numerals, punctuation marks and European accents. But to cater for more languages or typographic extras, you need a separate font, sometimes called an 'expert set'. The same usually happens with TrueType fonts. For example, a typeface with 'oldstyle' figures (which align with lower case rather than upper case) might also come with regular 'lining' figures. Fonts for high-quality text setting should have true 'small caps', which are reduced in height but match in weight (unlike capitals, which are scaled down). Some typefaces have optional 'swash caps', upper case letters with flourishes, or decorative ornaments.