Free morphemes like town, dog can appear with other lexemes (as in town hall or dog house) or they can stand alone, or "free". Allomorphs are variants of a morpheme, e.g. the plural marker in English is sometimes realized as /-z/, /-s/ or /- ɪz/.
Bound morphemes like "un-" appear only together with other morphemes to form a lexeme. Bound morphemes in general tend to be prefixes and suffixes. Unproductive, non-affix morphemes that exist only in bound form are known as "cranberry" morphemes, from the "cran" in that very word.
Inflectional morphemes modify a word's tense, number, aspect, and so on. (as in the dog morpheme if written with the plural marker morpheme s becomes dogs).
Derivational morphemes can be added to a word to create (derive) another word: the addition of "-ness" to "happy," for example, to give "happiness."