The phrases in question have to do with the Canon of Scripture. The term "that which is in part" refers to Scripture penned prior to the Incarnation; the term "that which is perfect" refers to the Scripture penned subsequent to the Incarnation. Protestants call the former the "Old Testament" and the latter the "New Testament"; but those designations are incorrect and misleading.
The fledgling Church, operating under the New Covenant, had need of information and instruction which had not yet been written and incorporated into the Canon of Scripture. There appear to have been four channels through which this vital information was supplied.
FIRST. The apostles constituted the primary source of information and instruction. The best example of this is Paul, whose letters form a large portion of the documents added to the existing Canon.
SECOND. Individuals having the spiritual gift of Tongues were given utterance by the Spirit of God. But the local assembly could not benefit from this utterance unless a translator was present while the utterance was in progress.
THIRD. Individuals having the spiritual gift of Prophecy were given utterance by the Spirit of God. In primary meaning, the terms prophet and cognates (prophecy, prophesy) have to do with "telling forth," in the sense of exhortation or rebuke. Many of the exhortations delivered by the prophets of old included the foretelling of events which then were yet to come. However, the Scripture records at least two instances in which, in the present era of the New Covenant, a prophet did foretell future events, Acts 11:28, Acts 21:10-11.
FOURTH. The spiritual gift of Knowledge appears to have been a rather comprehensive understanding of the Scripture, the insight being imparted apart from intensive study. Consider the plight of a new local assembly comprised of "Gentiles" having limited understanding of the Scripture, yet having immediate need to understand many things. Years of study of the Scripture were impossible. However, it would have been necessary that the gift of Knowledge be imparted to only one member of the assembly, who then could teach the other members of the assembly.
In the passage in view, I Corinthians chapter 13, Paul is correcting the Corinthians regarding their view of the spiritual gift of Tongues. The Corinthians assigned prestige to the gift, assuming that the gift was associated with spiritual status, and that the more one spoke in tongues, the more lofty his status.
Paul declares that the gift of Tongues, along with the gift of Prophecy and the gift of Knowledge, were temporary measures which would be discontinued once the Canon of Scripture was completed. The Greek term TELEIOS means mature, complete, without lack; the English term "perfect" has the same meaning, but English speakers typically misconstrue the Scriptural concept of perfection. Thus, upon completion of the Canon ("when that which is perfect is come"), the gifts of Tongues, Prophecy, and Knowledge ceased ("then that which is in part shall be done away").
Though there is no record regarding the actual date at Canon was completed, it is certain that no Scripture was penned subsequent to the death of the last surviving Apostle.