English th represents two different sounds, the sound of th in thin and the sound of th in then. The two sounds are quite rare in the world's languages: only Greek and Icelandic among the European languages have both. Although they are as different as f and v or s and z, there has never been any way of distinguishing them in English orthography (in the IPA they are written Θ and ð respectively). This isn't as bad as it seems, because in fact the second sound is used only in a minority of words, which basically fall into four categories:
- in native English words between vowels
- in native English words at the end where there used to be a vowel that has been lost, often with a silent -e that represents it
- at the beginning in closed-class words
- just before a final m with no vowel separating them
Examples of the first group are: bother, brethren (formerly bretheren), brother, either, farther, father, fathom, feather, further, gather, hither, leather, mother, neither, nether, other, rather, slither, smithereens, smithy, smother, swarthy, together, weather, (bell)wether, whether, (where)withal, wither, withershins (a variant of widdershins, a rare word meaning 'counter-clockwise') worthy, and their inflected and derived forms. Either appears on this list, but ether does not, because it is borrowed from Greek. Thither belongs to both this group and the third group.
Examples of the second group are: bathe, bequeath, betroth, blithe, breathe, clothe, lathe, lithe, loathe, scythe, seethe, smooth, soothe, teethe, tithe, withe, wreathe, writhe and their inflected and derived forms. The verb mouth (not the noun mouth) also belongs to this category.
Examples of the third group are: than, that, the, their(s), them, then, thence(forth), there (and compounds), these, they, this, those, (al)though, thus, thy(self). All these words belong to closed classes, grammatically similar groups of words that aren't easily added to English, such as pronouns, conjunctions, and articles. Open classes like nouns, verbs, and adjectives don't normally have this sound at the beginning of the word: thigh is a noun and has the first th sound, whereas thy is a pronoun and has the second.
Examples of the fourth group (the only ones I can find) are algorithm, logarithm, and rhythm.
There are very few minimal pairs for these two th sounds; that is, pairs of words which are distinguished only because one has the first and the other the second sound. Thigh and thy, mentioned above, form perhaps the best-known pair. For some English-speakers, thin and then are likewise a minimal pair, because they do not distinguish between short e and short i before m or n.
Earlier varieties of English (and perhaps in some dialects still) had no phonemic voicing contrasts in fricatives. Since then, all fricative phonemes save h have split (f and v, s and z, Θ and ð, and ʃ and ʒ) but to varying degrees, and even now there are not very many minimal pairs for any of the four.