Spirit of the Lord Upon David 1 Samuel 16:13 states that:


  So Samuel took the horn of oil and anointed him in the presence of his
  brothers, and from that day on the Spirit of the LORD came powerfully
  upon David.


and in 14:


  Now the Spirit of the LORD had departed from Saul


Is this a description of the holy spirit?

Thanks.
It is, indeed, presumed to be the Holy Spirit.  There is the idea that the Spirit did not indwell people in Old Testament times, but did "come upon" them.  The indwelling did not take place until after redemption has been accomplished with the death, burial and resurrection of Jesus.


  Or do you not know that your body is a temple of the Holy Spirit
  within you, whom you have from God? You are not your own  1
  Corinthians 6:19 ESV


A key question would be what Spirit could it be if it is not the Holy Spirit.  There really are no other legitimate candidates for that.  The word "spirit" is typically used of our own spirits and of the Holy Spirit.  Since this particular use of "spirit" is distinguished as that of the Lord (and not ours), the best conclusion appears to be that it is, in fact, a reference to the Holy Spirit.

It is interesting to note that David specifically prayed that God would not take the "Holy Spirit" from him after his sin with Urriah and Bathsheeba:


  Cast me not away from your presence, and take not your Holy Spirit
  from me.  Psalm 51:11 ESV