Ah ok, you asked about the actual kind of ranks, actually you can find them scrolling the filter for ranks in the ranked matches, or alternatively doing the same thing when you are creating a lobby =)
By the way Mokyu already answered you, for being more precise currently they are:
LR
F
F+
E-
E
E+
D-
D
D+
C-
C
C+
B-
B
B+
A-
A
A+
S-
S
S+
U
Topaz
Emerald
Ruby
Sapphire
Diamond
Any points for that list? And I am not doing it for bragging rights I am doing it because LR looks ugly and if I am going to upgrade I am going to go for something decent. I have no interest in bragging. Ever.
I can find a list made for a member of FSD, by the way this goes only from F to S:
F Rank-Given at the start (in DoA5 Vanilla)
F+ Rank-400
E- Rank-1,000
E Rank-1,800
E+ Rank-3,300
D- Rank-4,900
D Rank-6,600
D+ Rank-8,400
C- Rank-10,300
C Rank-12,300
C+ Rank-14,400
B- Rank-17,600
B Rank-21,200
B+ Rank-25,200
A- Rank-30,000
A Rank-34,900
A+ Rank-40,000
S- Rank-51,000
S Rank-63,000
S+ Rank-78,000
In Last Round they added the LR rank I believe specifically for all the people that don't play ranked matches but only lobbies. In this way they can show a more nice classification that is related to them playing the game ("LR") and not an "F" that make little sense in their case (how they don't wish for a real rank that indicate an - online - skill)
Finally notice that originally in DoA5 the ranks went from F to S+. During the DoA5U age, at the launch of the game they added the U(Ultimate) rank, then for celebrating a certain amount of downloads of the Core Fighter version they added Topaz, Emerald and Ruby in a first time, more recently for the same reason even Sapphire and Diamond. So in future more higher ranks could come, as this is already happened so far.
You are welcome =) I will try to look for infos about the other ranks. I can just say you that I believe that U is reached around 98,000 points.
And like always, remember that the online skill doesn't say anything about your real one offline (cause the not perfect online of the DoA5 series). So don't worry to much for wins or loses (or flame/rage mails), rather play rank very carefree