Unless your resume is written in Chinese, Japanese, Hebrew, or any other language that is not based on an European character set use ANSI. 

ANSI is the old.  Unicode is the new.  However, the technology to handle unicode is not so widespread.  There are still email clients and a host of disparate systems that still process data in ANSI.  All systems that process data in unicode are smart enough to process ANSI when they see it.  There are still too many systems based on ANSI that know absolutely nothing about unicode and are surely to treat your resume unpredictably.  Therefore, unless you absolutely know you will need a resume in unicode, you should select ANSI every time.

ANSI is the original means that PCs used to represent characters on the screen much like the characters in this sentence.  Its shortcoming is that ANSI only contains 256 unique characters.  ANSI cannot represent all of the characters in the world such as Chinese ideograms and Sumerian hieroglyphics.

Unicode is the new.  It contains 65,535 unique characters which is enough to represent just about every character in every language of the world.  However, until every system in the world reads unicode your resume could wind up in a black hole.  Unless you are instructed to do otherwise, stick with ANSI.