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Definition:

To sanctify is to set apart or to make holy. Sanctification is the process of being made holy.

In the Old Testament, certain people and things were sanctified, or set apart, for service to God.
The New Testament teaches that God sanctifies people who believe in Jesus. That is, he makes them holy and sets them apart to serve him.
Believers in Jesus are also commanded to sanctify themselves to God, to be holy in everything they do.

Depending on the context, the term "sanctify" can be translated as "set apart" or "make holy" or "purify."
When people sanctify themselves, they purify themselves and dedicate themselves to God's service. Often the word "consecrate" is used in the Bible with this meaning.
When its meaning is "consecrate," this term could be translated as "dedicate someone (or something) to God's service."
Depending on the context, the phrase "your sanctification" could be translated as "making you holy" or "setting you apart (for God)" or "what makes you holy."