Mr. Embry is Finch's guidance counselor. He seems to genuinely care about Finch's wellbeing, but he's super ineffective. (That's partially Finch's fault, because he deletes Mr. Embry's voicemails to his mother.)
He's not totally useless, though. In Mr. Embry's last session with Finch, he mentions bipolar disorder—how long had he suspected?—but by then, it's too late. Later, he tells Violet,
"Could I have done more? Possibly. Yes. We can always do more. It's a tough question to answer, and ultimately, a pointless one to ask." (55.16)
Maybe if Finch had trusted him more, Mr. Embry could have helped.