Dfyjkt

All in all the same as the answer by Steven B. Segletes, but expandable. Also almost everything as contents should be fine (except the really unlikely endabr@).

documentclassarticle

makeatletter
newcommandabr[1]
 %
 abr@#1endabr@
 
defabr@#1#2endabr@
 %
 #1.%
 ifrelaxdetokenize#2relax
 @%
 expandafter@gobble
 else
 ,%
 expandafter@firstofone
 fi
 abr@#2endabr@%
 
makeatother

begindocument
Single letter:
abreg

Multi letter: 
abrthe
enddocument

Here is the simplest form of my original approach. It can be placed in an edef. Its only drawback is that it can blow the stack if the argument is too long (maybe 256 characters??)

documentclassarticle
newcommandabr[1]abraux#1relaxrelax
defabraux#1#2relax#1.ifxrelax#2relax@else,abraux#2relaxfi
begindocument
Here is abrOED abbreviation.

Here is abrXO abbreviation.
enddocument

If that really were an issue, here is an alternative that doesn't have that problem.

There seems to be a misunderstanding that this definition of abr cannot be placed into an edef. It can. Naturally, the expansion is not necessarily pretty, but it will yield the proper typesetting. The only proviso is that @MyOwnMacro is not used elsewhere in your document.

documentclassarticle
usepackage[T1]fontenc
makeatletter
let@MyOwnMacrorelax
newcommandabr[1]abraux#1relaxrelax
defabraux#1#2relax%
 #1.ifxrelax#2relaxdef@MyOwnMacro@elsedef@MyOwnMacro,abraux#2relaxfi%
 @MyOwnMacro%

makeatother
begindocument
Here is abrOED abbreviation.

Here is abrXO abbreviation.

Can be edef'ed:

edeftmpabrOED detokenizeexpandaftertmp

expands to tmp
enddocument

Here's a LuaLaTeX-based solution.

• Cases such as abrOED or abrIMF work just as expected. If the acronym contains both uppercase and lowercase letters, dots are inserted only before the uppercase letters in the interior of the acronym. E.g., abrMSc generates M.,Sc., and abrPhD generates Ph.,D..




• It can handle mixed-case acronyms such as "PhD" directly -- no need to write abrPhD.




• If a "slovenly abbreviation" ends a sentence, one should place the "." punctuation mark inside the argument of abr. The code takes care to insert a @ "space factor* directive before the final . character. This, in turn, informs LaTeX that that . character should be treated as ending a sentence.




• The code returns nothing if the argument of abr is either empty or expands to return nothing. E.g., defttt /abrttt/ returns //. If the code encounters non-letter characters -- say, ( and ) -- no periods are inserted before or after them.




• The code is expandable in the sense that abr can be included in the argument of an edef directive.

documentclassarticle
usepackageluacode % for 'luacode' environment

%% Lua-side code:
beginluacode

function abr ( s )
 n = string.len ( s )
 -- Do nothing unless "s" is non-empty.
 if n>0 then 
 s_mod = "" -- initialize the string

 -- Process the first n-1 characters in "s"
 for i=1, n-1 do
 s12 = string.sub ( s , i, i+1 )
 s1 = string.sub ( s12, 1, 1 )
 if string.match ( s12 , "%a%u" ) then
 s_mod = s_mod .. s1 .. ".\,"
 else
 s_mod = s_mod .. s1
 end
 end

 -- Process the final character in "s"
 s_n = string.sub ( s , n)
 if string.match (s_n, "%.") then -- "." char.
 s_mod = s_mod .. "\@." 
 elseif string.match (s_n, "%l") then -- lowercase letter
 s_mod = s_mod .. s_n .. ".\hbox"
 elseif string.match (s_n, "%u") then -- uppercase letter
 s_mod = s_mod .. s_n .. "."
 else -- Any other character:
 s_mod = s_mod .. s_n -- don't add anything after 's_n'
 end

 -- Print the modified string
 tex.sprint ( s_mod )
 end
end

endluacode

%% LaTeX-side code: macro that calls the Lua function
newcommandabr[2]directluaabr("#1")

begindocument
abrOED, abrPhD, abrDPhil, abrMSc, abr()

smallskip
% Two calls to "abr" with an empty argument (upon expansion):
.abr. quad 
defttt .abrttt. 

bigskip
edeftmpabrMA detokenizeexpandaftertmp

edeftmpabrMA. detokenizeexpandaftertmp

expands to: tmp

smallskip
edeftmpabrMSc detokenizeexpandaftertmp

edeftmpabrMSc. detokenizeexpandaftertmp

expands to: tmp

bigskip
Some tests of spacing after punctuation marks:

smallskip
a abrPhD candidate --- good

a Ph.,D. candidate --- just to verify

smallskip
an abrMSc candidate --- good

an M.,Sc. candidate --- just to verify

smallskip
She has a abrPhD. So do I. --- good

She has a Ph.,D@. So do I. --- just to verify

smallskip
He has an abrMSc. So do I. --- good

He has an M.,Sc. So do I. --- just to verify

smallskip
Does he have an abrMSc.? Really?! --- good

Does he have an M.,Sc.? Really?! --- just to verify
enddocument 

Let TeX do the recursion:

documentclassarticle
usepackagexparse
usepackageetoolbox

robustify, % just in order it doesn't expand in edef

ExplSyntaxOn
NewExpandableDocumentCommandabrm
 
 tl_map_function:fN tl_range:nnn #1 1 -2 __toothrot_abr:n
 tl_range:nnn #1 -1 -1 .
 
cs_generate_variant:Nn tl_map_function:nN f 
cs_new:Nn __toothrot_abr:n #1., 
ExplSyntaxOff

begindocument

abrOED

abrPhD

edeftestabrOED
textttmeaningtest

edeftestabr
textttmeaningtest

enddocument

If a part of the argument is braced, it is considered as a single item.

One might check whether the argument is empty in order to print nothing at all, but it doesn't seem so important a feature.

With tl_range:nnn #1 1 -2 we extract all items but the last; tl_range:nnn #1 -1 -1 extracts the last item.

documentclassarticle
usepackagexinttools
newcommandabr[1]xintListWithSep.,#1.
begindocument
I've seen it in the abrOED.

I got my abrPhD.
enddocument

Updated (à la Mico, but without LuaLaTeX)

The syntax here is to use abrPhD. for example at end of a sentence, and abrPhD if not at end of a sentence.

documentclassarticle
usepackagexinttools
makeatletter
newcommandabr[1]
 expandafter@gobbletworomannumeral0xintapplyunbracedabr@aux#1.@
defabr@sep.,
defabr@aux#1if.#1expandafterabr@end
 else
 if1ifnum`#1<`A 0fiifnum`#1>`Z 0fi1%
 expandafterexpandafterexpandafterabr@sep
 fi
 fi#1%
defabr@end. abr@@end
defabr@@end.@@.
makeatother

begindocument%ttfamily
I've seen it in the abrOED, and if located at end of a sentence
just insert a dot in the verb|abr| argument: abrOED. It ended a
sentence and in non-French spacing mode, TeX inserted the extra
space.

textttWe can see it better with monospace font: abrOED. See?

I got my abrPhD and even my abrPhilD, leniency ruled
in those days.

textttThe dots are added in a smart way: abrAaaaBbbbCccc. But it is
assumed that the first letter is abrUppercased. That's it.

textttNotice that neitger abrAaaa nor abrAaA trigger an end of
 sentence spacing after the dot, which is abrGood. Isn't it?
enddocument

Again updated, for automatic end of sentence dot detection after abbreviation

Here an end of sentence dot will be detected automatically.

Of course we can't use @ifnextchar for that, as it swallows spaces.

I added some comments about expandability, which seems to have drawn great attention in other answers :).

documentclassarticle
usepackageshortvrbMakeShortVerb
usepackagexinttools

makeatletter
protecteddefabrsep.,% maybe redefined even after edeffooabrDPhil...
protecteddefabrendfuturelet@let@tokenabr@end
defabr@endifx.@let@token@else.@fi
newcommandabr[1]
 expandafter@gobbleromannumeral0xintapplyunbracedabr@aux#1abrend
defabr@aux#1if1ifnum`#1<`A 0fiifnum`#1>`Z 0fi1%
 expandafterabr@sep
 fi#1%
defabr@sep abrsep
makeatother

begindocument%ttfamily
I've seen it in the abrOED, and if located at end of a sentence
it will detect it automatically: abrOED. There was no double dot.
Besides, TeX applied its end of sentence extra space.

textttWe can see it better with monospace font: abrOED. See?

textttWe can see it better with monospace font: abrOED, See?

I got a abrMSc, a abrPhD and even a abrDPhil. Leniency ruled
in those days. 

The abbreviation dots are added in a smart way, after the last
 lowercase letter following an uppercase letter:
 abrAaaaBbbbCccc. But it is emphassumed that the first letter is
 abrUppercased. That's it.

textttNotice that neither abrDPhil nor abrPhilD get TeX to
 consider the inserted final dot as signaling an end of
 sentence spacing after the dot, which is abrGood. Isn't it?

About expandability, the correct way for LaTeX2e's users would be to use
|protected@edef|, not a naked |edef|; although nowadays some
LaTeX2e users have heard about |edef|, they might not know
about |protected@edef|, which requires a cumbersome extra
|makeatletter| for its usage. Anyway, none of that is described in
textscLamport book, so I wonder if LaTeX2e users are really
emphallowed into using |edef| to start with.

But as it seems they know about |edef|, we as macro programmers need
better to use the e-TeX's |protected| prefix and not the LaTeX2e
|DeclareRobustCommand|, because users will not do |protected@edef|.

This is what I have done here for a macro |abrsep| (why haven't we all
used |abbr| by the way?) which is deliberately |protected|,
allowing it to be redefined at location of use, long after some macro
will have been defined via |edeffooabrABCDEFGH|.

edeffooabrABCDEFGHtextttstringfoo is meaningfoo

edeffooabrAaaBccCcctextttstringfoo is meaningfoo

The |abrend| is also |protected|, anyway as its expansion will be
context dependent (it detects if a dot follows), it had to not expand
in the |edef|.
enddocument

Notice that my proposals v2 and v3 will work only with ascii uppercase letters, no diacritics.

documentclassarticle
newcommandabr[1]abraux#1..
defabraux#1#2#3%
 #1%
 ifnum`#2>91relax% we have a lowercase letter following
 defnextabraux#2#3%
 else
 .ifx.#2defnext@else,defnextabraux#2#3fi
 fi
 next

begindocument
 Here is abrOED abbreviation.

 Here is abrXO abbreviation.

 Here is abrXOOED abbreviation.

 abrMSc 

 abrDPhil
enddocument