What exactly did Pope Gregory the Great mean by "Universal Bishop?"

Does it not answer itself? He said any man who calls himself this... Rather than God calling him it.

Also later Gregory says a christian council gave that name to the succesors of Peter but he wouldn't even use it even though aloud because of the impression it would make.

Refer, please, to Matthew 20:26. Christ tells His apostles that whoever would be great in the Kingdom of Heaven must be your servant.

Every group must have a head that directs it or it soon disintegrates. Every corporation must have a CEO or some such position.

In the Catholic Church, yes, the Bishop of Rome is the head of the College of Bishops. His position, however, is not to Lord it over the other bishops, but rather to assure that each diocese is given an equal hearing as to its problems and needs. His ultimate assignment by God in the power of the Holy Spirit is to assure that Christ's teachings are passed on correctly from generation to generation so that all people hear the truth of God's Word correctly interpreted and lived out in the Church's actions.

I hope that helps. Believe me, no man in his right mind truly wishes to be Pope. It is a frightful position to hold, one that allows for little time to oneself and little peace. Picture a father with multiple children, all of them wanting individual attention.

How does a Catholic layman reconcile these two contexts?

If we understand Pope St. Gregory I ("the Great") as denying that the pope is the "universal bishop" and taught that the Bishop of Rome has no authority over any other bishop, then the layman is left with more questions than answers. For example:

1. How is he Saint and Doctor of the Church?
2. With his denial, how is it he remained Pope till his death? Why did he not resign, abdicate or abolish the papacy?

Thus there must necessarily be a context for his statement and I found an explanation here:

Pope Gregory the Great and the "Universal Bishop" Controversy: Was the Pope denying his own Papal Authority? (Debunking a popular Protestant myth).

The following is the reason the article gives for the Pope's condemnation:

What Gregory condemned was the expropriation of the title Universal Bishop by Bishop John the Faster, the patriarch of Constantinople, who proclaimed himself Universal Bishop at the Synod of Constantinople in 588. Gregory condemned the patriarch's act because universal jurisdiction applies solely to the pope.

Users are encouraged to read the entire article.

Further reading:

• Quick Questions (1992) | New Advent.
• Gregory to Mauricius Augustus | Book VII, Letter 33 | New Advent.

In your question it would be very helpful to have a citation as to where in the writings of Gregory the quote is from. I followed the link provided and there is no citation there for the quote either. As a matter of fact, you can google the first sentence of the quote and find a few websites that also have the quote but leave it uncited. Knowing its context is crucial to understanding what it is saying. As a note of caution, taking Papal quotations out of context seems to be a cottage industry on the web.

The quote and its full context can be found here: Gregory to Mauricius Augustus | Book VII, Letter 33 | New Advent.

And an explanation of what Gregory was addressing can be found here: Quick Questions (1992) | New Advent.

What follows below is taken from the New Advent article.

"What Gregory condemned was the expropriation of the title Universal Bishop by Bishop John the Faster, the patriarch of Constantinople, who proclaimed himself Universal Bishop at the Synod of Constantinople in 588. Gregory condemned the patriarch's act because universal jurisdiction applies solely to the pope. . . Predictably, anti-Catholics neglect to inform their audiences that the context of these statements makes it clear that Gregory was not making these statements in regard to himself or to any other pope. He believed the Bishop of Rome has primacy of jurisdiction over all other bishops."

The two contexts are not in contradiction when they are properly understood.

Secondly, the second quote comes from a Church Council. Church Councils can produce dogmatic teaching. What is written by Popes can sometimes be considered Church teaching, but not dogmatic teaching. What is written in letters by Popes is not something that could not change. As to the status of what is quoted by Gregory, it seems to me that Gregory it not really trying to make a Church teaching, but is telling a particular person what he thinks.