'외롭다' is an example of what is called a 'bieup irregular verb' (비읍 불규칙동사, in Korean). Most of the them are adjectival and the reason for this is historical. It used to be that many adjectives were derived from verbs by adding an infix consisting of a letter called a 순경음 비읍 and either an 아래 아 or an '으' for the vowel. Neither the 수경음 비읍 nor the 아래 아 exist in modern Korean. Phonetically, 순경음 비읍, which was written as a 'ㅂ' with an 'ㅇ' under it, was a bilabial fricative. Think of a 'v' sound in English but instead of putting your lower lip to your upper teeth, just put your lips together like you were going to pronounce 'p'.
그리다 (v.t. meaning 'to miss') -> 그립다 (modern adjective meaning 'to be missed' or 'to be longed for') -> from which is further derived '그리워하다' (v.t. also meaning 'to miss').
It's because these verb stems historically did not actually end in 'ㅂ' that the modern forms tend to remain irregular. Note that not all verb stems ending in 'ㅂ' are irregular, though any verb ending in '-롭다' will be irregular.
As for how to conjugate them, the rule is simple. When a 'ㅂ' irregular verb stem is followed by a vowel, the 'ㅂ' weakens to a glide (i.e., a 'w') which then merges with the following vowel.
So when you're adding the conjunctive '-어/아', you get either '-워' (weo) or '-와' (wa).
When a 'ㅂ' irregular verb stem is followed immediate by an '-으', the '으' drops (it's the weakest vowel in Korean and frequently drops). For example, take 덥다 (to be hot):
덥 + 어 -> 더워
덥 + 어서 -> 더워서
덥 + 었어 -> 더웠어
덥 + 으니까 -> 더우니까 (notice how the '으' drops away and you are left with '우')
덥 + 으면 -> 더우면 (same as above)
덥 + 지 -> 덥지 (no change when followed by a consonant)
덥 + 습니다 -> 덥습니다 (-ㅂ니다 after a vowel but -습니다 after a consonant; since the 'ㅂ' verb ending only weakens when followed by a vowel, you'd use the consonant version here: -습니다).
Similarly for '외롭다', you get 외로워, 오로웠어, 외로우니(까), 외로우면, 외롭습니다, 외로웠습니다, 외롭지, etc.
If it helps you to remember it, learn the following cultural reference: "외로워도 슬퍼도 나는 안 울어..." (even though I'm lonely or sad, I don't cry).