Does the start equal the end?

Python 3, 23 bytes

s=input()
s[0]!=s[-1]<e

Output is via exit code, so 0 (success) is truthy and 1 (failure) is falsy. If this is acceptable, a byte can be saved.

Try it online!

How it works

First of all, if s is an empty string, s[0] will raise an IndexError, causing the program to fail.

For non-empty s, if the first and last characters are equal, s[0]!=s[-1] will evaluate to False, so the program exits cleanly and immediately.

Finally, if the characters are different, s[0]!=s[-1] will evaluate to True, causing the compairson s[-1]<e to be performed. Since e is undefined, that raises a NameError.

If backwards compatibility with Python 2 is not desired,

s[0]!=s[-1]<3

works as well, since comparing a string with an integer raises a TypeError.

JavaScript, 19 bytes

a=>a.endsWith(a[0])

Retina, 13 12 bytes

^(.)(.*\1)?$

Try it online! Includes test suite. Edit: Saved 1 byte thanks to @Kobi.