Take the three posts by Keltoi, Woodrow and Joe and add this one to it, and I think you'll have more information than you can use to answer your questions.
The New Testament church did not actually celebrate Christ's birth as a holy day, just his resurrection. But in later centuries the church decided that they wanted to celebrate the date of his birth as well. They had no idea what that date actually was. Many theories were advanced from sometime in the spring lambing season to, because of a possible celestial event in the estimated year, sometime in October (I was born that month being the reason I prefer it) and, yes, a few even argued for sometime in December. But as no one really knew, and no particularly compelling date could be chosen for any one date over another, the powers that be in the church at the time ultimately had to just make an arbitrary choice of some date, it could have been any date, and they choose December 25. As I am sure you would have heard from those who like to ridicule Christmas, the reason for choosing it was because there happened to be a pagan celebration of the winter solstice around that same time. In the mind of the leaders of the church in that day, the best way to encourage the people to give up their pagan celebration was to given them a Christian one to substitute in its place, so rather than celebrating the "rebirth" of the sun, they were instead encouraged to celebrate the birth of the Son of God. Once that date was set on the calendar, it as was explained above, became celebrated on a given date like any other birthday celebration.
Oh, and by the way, even this December 25 date isn't universal. Not all of the church chose to celebrate Jesus' birth on December 25. Some parts of the church, for reasons I cannot detail, opted instead to celebrate Jesus' birth on January 6. In the majority of the church, January 6 is recognized the day that the Maji arrived to visit the infant (or maybe better understood as 2-year-old toddler) Jesus and is called Epiphany. Both dates have been used for centuries with Christians aware of them and generally not arguing about which is right, because it is understood that no one really knows. We just want to celebrate and one day is as good as another. The result for many centuries (until, in commercializing Christmas, people started thinking of Christmas as a season started following Mr. Macy's Thanksgiving day parade and ended with tearing packing up the tree the day after Christmas) was a 12 day celebration from December 25 to January 6 called the 12 days of Christmas -- and, if you ever sang that song, now you know the rest of the story.
Easter, again as noted in posts above, is tied to the Jewish celebration of passover. Because the Jewish calendar is not linked to either the Gregorian or Julian calendars (both intended to be purely solar calendars), but is a variation of a lunar calendar within a solar year the date moves on the Gregorian solar calendar. Perhaps this can best be explained with a little story.
Passover, or Pesach as most Jews would refer to it, is the most well known of all the Jewish holidays. Pesach begins on the 15th day of the Jewish month of Nissan, but as you are well aware, that date moves on the Gregorian calendar that is used in everyday 21st century life. As the most commonly known of all Jewish holidays, it is even honored by non-observant Jews. So, confused by this "moving" of the date of Pesach, a man visits his local rabbi.
Rabbi: Samuel, how go it is to see you today. We've been missing you.
Samuel: Thank-you, Rabbi. I have been lax in my duties to Torah. But I wish to make amend.
Rabbi: Very good, Samuel. Then we shall see you at Pesach.
Samuel: Most certainly, Rabbi. Only, Rabbi, I must ask for one thing.
Rabbi: Of course, Samuel, what is it?
Samuel: When is Pesach this year?
Rabbi: The same as it is every year, Samuel; the 15th of Nissan.
This same conversation could happen with respect to Easter. The date appears to move, but in reality it is fixed. It is always the first Sunday, after the first full moon, after the March equinox. If you want more detailed information on how that is determined and the all the little quirks of history involved you can read a rather detailed telling of that
here.