Generally speaking, "the anointing" refers to the "baptism of the Holy Spirit." More particularly, someone who is "anointed" or "the Lord's anointed" is believed to have some greater share of the Spirit and has been gifted by God with a special role in the church.
Do you have something or someone specific in mind?
Not really. I remember this guy attending a conference several years ago who kept using that word. Seem like he was trying to get his theology right, but old ways are hard to lose.
Plus there hasn't been much traffic on this forum lately,
"having been firmly rooted and now being built up in Him and established in your faith . . ."Colossians 2:7