It is an auxiliary verb with the the same meaning as "must". Here is what dictionary.com says:
auxiliary verb
2. Informal . must; have got (followed by an infinitive).
So, in your example, the meaning is "I must get..." Note, it is used with "have" which again is an auxiliary meaning "must", and I can't think of an example of where you could use "get" in this way without "have". But perhaps I am wrong about that. In this case, using both adds an emphasis to the importance of the "getting".
Note, as the dictionary says it is pretty informal, and should not be used in any type of formal communication.
The OED has this usage back to 1849 so it's been around a while. It says that it comes from omitting have and is "colloquial":
b. The pa. pple. [past participle] is also used colloq. with omission of (I) have. Cf. gotcha n., gotta v.
1849 Knickerbocker 34 12 They got no principles. They got no platform to stand onto.
1857 Quinland I. 1 Got an hour to spare—thought I'd just run in and see what you were all about.
1884 ‘M. Twain’ Adventures Huckleberry Finn xxxviii. 325 We got to dig in like all git-out.
1887 M. E. Wilkins Humble Romance 370 What you got there, grandma?
1911 R. D. Saunders Col. Todhunter i. 11 Oh, of course, you got to laugh at me.
1911 J. F. Wilson Land Claimers ix. 118 But I got several plans, and I need ye.
1941 P. F. Webster & D. Ellington (title of song) I got it bad and that ain't good.
1967 L. White Crimshaw Memorandum (1968) v. 93 Gawd knows I got enough problems.
EDIT: I don't have evidence, so I didn't originally include it in my answer, but my suspicion is that:
In US informal registers, got seems to have been re-interpreted as a present-tense verb form just meaning "have, possess". It sure behaves that way. It's homophonous with, but not identical to, the past tense of get. Historically it seems to have been a resultative construction, but it acts like a normal verb now.
The only issue is if so, then the verb's defective in the 3sg: both "he/she/it got" and "he/she/it gots" are highly marked and are just avoided in most dialects. In response to Betty's inquiry, I'm not sure sure if people just say "he/she/it's got" or if we reword to avoid the issue.
Again, I got no evidence; it's just a pet theory for now.
Best Answer
To answer the original question, it's Present tense, and the verb construction is Perfect. Together they're often called "Present Perfect", which is a tense in Latin. but only a construction in English.
Yes, have got is an idiom; but that explains nothing except its irregularity, of course. And it's a different idiom in UK English than it is in American English, where it contrasts with have gotten. Like most idioms, where it came from is a long tortuous story.
Get means come to be or come to have, as in
or
(I've always been bemused that in German the verb bekommen means receive but not become.)
In the case of have, especially, if one comments on the acquisition of something, the implicature is that one still has it -- otherwise, one would say something different. So the present perfect of get naturally implicates the present of "have", leading to the equivalence of have got and have.
The Present Perfect construction uses the auxiliary verb have/has, plus the past participle of the matrix verb:
The past participle of get is got or gotten in the US; UK mileage may vary. There is a principled distinction between the two, since get -- as the inchoative form of both be and have -- is itself an auxiliary, and got has come to have its own usages in American English, leaving the simple Past Participle slot to be filled by gotten.
As McCawley points out, one of the functions of the Perfect is to report past actions still relevant in the present; thus,
reports a past event (catching the cold) which is still relevant (having the cold), and, since pragmatically what we're interested in is the present state, I've got a cold is used more often to warn people to duck when I sneeze than to comment on the events of the past week.
But wait – there's more. Both be and have are already auxiliary verbs, and are used in many constructions, like Passive or Perfect. Since get can implicate be and have in some cases, it's been generalized to substitute in others, where their use is grammatical instead of meaningful, like the so-called Get-Passive
or in the periphrastic modal have to meaning must
(frequently spelled gotta, because the /v/ or /z/ in /ðevgaɾə/ /hizgaɾə/ is usually inaudible)
or simply, wherever one might use have
Quite frequently children generalize this equivalence to produce sentences like
in effect, inventing a new verb because the old one has worn out.
Got all that?