But in general, it means one tells the other what to do, and/or does it to the other.
As for the word "own", I'd be very concerned about that. Seems like the person is trying to inveigle you into being financially responsible for them (Which, is actually a form of BDSM that is practiced by some.)
... some more context.
BDSM can range from simple role play (Doms take charge, Subs comply) to heavy use of toys/tools/outfits (restraints, clamps, whips, spankers, leather suits, racks, pain devices, etc.). This is not strictly a Sugar Bowl play. There's lots of material on the net. But essential to know who is the Dom, the Sub, and if either or both can/will Switch. The hardware is really not my thing. But I do like a submissive woman in general.
DDLG = Dear Daddy Little Gal. The idea is deep Daddy/Daughter role play and could include pet names, schoolgirl/teacher costumes, presents, spanking (when she's a "brat") and rewards. Tread carefully here. If she's really into it, cool. But be careful NOT to extend the fantasy to include "Little Gal's" that are under age.
"I want a daddy to own me" could be anything or nothing. Maybe it's the ultimate Sub who will give herself without qualification. Or maybe she wants to move-in with you, or have you pay her rent and everything else. My suggestion: Make that your opening question on your message to her. "Tell me how you want Daddy to own you?"
In fact any of these can be a good opener for your message. It shows you read the profile and you are interested in her ideas and desires.
Life is good
The Cat 
Since I haven't met anyone like that, but DDLG might not involve sex!!! So not interested.
You’d be wrong there. Imagine it’s sex with some specific prolonged foreplay.
Only thing HC didn’t say is it’s a type of BDSM.
And it’s stands for “Daddy dom/ little girl” (not “dear daddy”). I think the confusion is there are a few Reddit posts that start with “Dear Daddy” letter from subs perspective.
As for the dom and sub thing. As others have said you can take it to any level you want. As simple as just being dominant in bed. I'm a dom and I don't really do the heavy stuff much. I don't mind it, but it's too much work honestly.
Example of my typical sex attitude. When my SB is starting to get undressed, I tell her I want to watch and go slower. I direct her. Then in bed I tell her what to do, so when I'm playing with her clit I ask her "are you going to cum for me", tell daddy you're going to cum for him. Then I pretend choke her a little and say "I didn't hear you". When she replies in the affirmative I go down on her to make her cum. That sort of thing. Dominance.
As far as own her, that's honestly, to me, a fantasy of something she has in mind or you both come up with together. Don't be afraid, outside the bedroom, to talk about it. For me own would be something like heavy dominance. Such as, I'm inside her and come up with something she has to answer no to. Then I would grab some of her pussy juice and starting getting ready for anal, not taking no for an answer (remember she answered no) and tell her daddy's going to punish you. Then starting fucking her in the ass.
Now, when she says no or protests you gotta make sure it's within the role play which means set up a safe word and go over it before each session so you both remember.
As a side note (I love verbal so it may just be me) some things translate to being super hot during sex and not so much after I nut. I love telling my babies 'you better clean daddy off with your mouth after I cum'. They oblige without me asking, but as you all know, most of the time, after you nut you don't want her sucking on you for at least a few minutes.
