Which sentence is grammatically correct? "That's her on the left." or
"That's she on the left."

One is gramatically correct, but sounds stilted.  You are more likely
to hear the gramatically incorrect one.

Questions like this one always remind me of an old joke:

[A knock on the door]
PERSON INSIDE THE HOUSE: Who is it?
PERSON WHO KNOCKED: It is I.
PERSON INSIDE THE HOUSE: Oh, the English teacher.

nelson is correct, here's an explanation of the reasoning:

(near the bottom)
http://www.nykola.com/archives/000615.html
I was also curious about "This is she / this is her." I was taught to
use the nominative case ("this is he"), but as I'm always interested
in improving my written English, I looked around the web for guidance.
I found this:

>>> The version that you use, "This is she," is what linguists call
"prescriptive grammar." This means that, rather than following the
native grammar of English, it follows rules that have been imposed on
English from outside (prescribed). In this case, the rule is based on
Classical Latin. In Latin, two noun phrases connected by a form of "to
be" are both given in the nominative case (the subject form). Thus
Latin speakers would use their version of "she" (nominative) rather
than "her" (accusative or dative). In English, however, the rule is
different. English native grammar allows only one noun phrase to have
subject case. All other nouns in the sentence are given in the object
case. Thus, the native English grammar produces the sentence "this is
her" ("this" is the subject and "her" is the object case).

These 2 links show how confusing this is, two grammer sites say the
exact opposite things:

http://www.press.uchicago.edu/Misc/Chicago/cmosfaq/cmosfaq.Pronouns.html
http://www.suite101.com/lesson.cfm/17951/1305/4