Any theory which explains a given subject-matter in terms of appearances, without needing to postulate anything else (see also reductionism), much as facts about the average man are reduced to facts about ordinary men. The most notable 19th century phenomenalist was John Stuart Mill (1806-1873).
Phenomenalists in the 20th century (for example, Alfred Jules Ayer (1910-1989)) usually take a more linguistic approach, reducing sentences about one kind of thing (in particular, material objects) to sentences about sense-data (things allegedly given to us immediately in sensations and so on). This approach suits logical positivism because of the difficulties of directly verifying sentences about material objects.
The difference between phenomenalism and subjective idealism can be seen by comparing two interpretations of how George Berkeley (1685-1753) accounts for ideas of unperceived material objects. On the idealist (and usual) interpretation such ideas exist in the mind of God. On the phenomenalist interpretation they do not exist at all, but God ensures we have the relevant ideas whenever (in common sense terms) objects come back into view. No successful phenomenalist analysis of material objects, however, has ever been completed.
J D Mabbott, 'The Place of God in Berkeley's Philosophy', Locke and Berkeley, C B Martin and D M Armstrong, eds