Well, that is really a hard question, so I will show you the the meaning of '나다' from my Korean-English dictionary for you. 나다 ①『태어나다』 be born; come into the world; come into being [existence]. ②『자라다』 grow;『돋아나다』 sprout; bud (out); come out. ③ (a)『발생‧일어나다』 happen; occur; take place; break out (전쟁‧화재 따위가); arise; have. (b)『생기다』 produce 「good results」; make [get] 「a profit」; yield 「profit」; generate 「heat」; cause 「damage」; come 「out of, from」; accrue 「from」. ④『소리가』 sound; come out [forth]; make a sound [noise]. ⑤『병‧증상 따위가』 become; fall; get; have. ⑥『명성‧소문 따위가』 acquire; get abroad; circulate 「among」. ⑦ (a)『흥미‧능률이』 grow to be; get. (b)『기분‧생각이』 occur; come [flush] across [into] one’s mind. ⑧『흘러나오다』 come out; gush out [forth]; flow [run] out. ⑨『생산되다』 be produced; be yield-ed; be raised [grown]; be found. ⑩ (a)『티가』 have an air [a look, a style]; look like. (b)『맛‧냄새가』 smell; taste. ⑪『신문 등에』 appear [come out] 「in」; go into 「a newspaper」. ⑫『늘다』 gain 「in」; gather. ⑬『결과 따위』 be forthcoming;『결과로서』 come out as a result; turn out (to be); turn up (as). ⑭『잘생기다』 be handsome; be good= looking; 『탁월하다』 be great; be extra-ordinary. ⑮『눈 밖에』 leave; go [get] out. ⑯ 『나오다』 appear 「on the market」; come on 「to the market」. ⑰ (a)『뚫리다』 be made; be open(ed). (b)『비다』 open up; be vacated; become vacant. (c) 『방향으로』 open on. ⑱ 『계절을 지내다』 go [get] through; pass 「a season」. ⑲ 『되다』 ⑳ 『흠 따위가 생기다』 flaw; crack. Isn't this word amazing? 나다 has really lots of meanings and we just know the meaning of the word by using the word everyday. Anyway, -었-, 았- and -났- make the word(나다) past tense. Here is an example. 일어나다 (infinitive or present simple) - 일어났다 (past/past perfect/present perfect/present perfect continuous)

you are asking a very useful question but I don't think I can provide a satisfatory answer. What I see over the years is that this 나 part encapsulates something really abstract but more or less associates with the concept of "coming forth" or "coming out". It has its counterpart - 내 which just changes the agent of action from itself (intransitive) to someone else to initiate the action (transitive). An example 티 나다 & 티 내다. When someone sees you and senses anger they might ask you "why are you so pissed today?" you could say " 티 나?" it means "does it show? (my face)". But if you intentionally try to show an angry face to someone else, you could use "티 내" essentially saying you make a face ( you will it to happen). I suggest you just learn the vocabulary as times goes you will get a feel for it. We can quote thousands of examples but there's always some cases where they just don't fit the expectation.