Today these are mostly frozen forms that behave like attributives (連体詞, see my answer here). Verbs in this class include 確たる (sure, certain) and 堂々たる (majestic). It's really not such a big class.
Originally, they were descriptive nouns or descriptive verbs that were followed by the auxiliary verb たり (itself a combination of the particle と and the verb あり, to be). So an expression like 確とあり (
kaku to ari), became 確たり (
kakutari), which could then be conjugated as if it were a verb.
Now, the forms ending in the vowel
i are all the conclusive forms (終止形), which were used in classical Japanese for verbs in the end of the sentence. Verbs that came before nouns used the familiar attributive forms (連体形), usually ending in
u, which are now used indiscriminately both in the end of the sentence and before nouns - so 確たり became 確たる in Modern Japanese.
Besides ～たる adjectives, there are also ～なる adjectives in Modern Japanese. Their history is quite similar: they used to be descriptive nouns or verbs followed by the old copula なり (like the auxiliary verb たり, the old copula also ultimately comes from a particle + verb combination, this time に + あり). Modern examples of such verbs include 単なる (mere, simple) or 聖なる (holy).