Do churches host distinctive, special services on Christmas, Christmas eve, and Easter? I have read before that some (nominal) Christians only attend worship services on Christmas and Easter. Are there anything special about those two holidays that compel them to attend church during Christmas and Easter? What do the churches do exactly that might convince people to attend church on only those days?

I know that stores hold big sales on those days, which may attract people to go shopping and buy merchandise.
Typically, yes, most denominations how special services on Christmas eve, and some, possibly, on Christmas day.  (More details coming.)

In Catholic Churches, there is the Midnight Mass.  There's also Christmas Vespers for several denominations including the Catholic Church, Eastern Orthodox, Lutheran, and Anglican Churches.

The LDS Church has some unique Christmas traditions, including visting Templa Square at Christmas, and watching the Church's annual "First Presidency Christmas Devotional" which is broadcast from the Conference Center (at Temple Square) to church buildings worldwide.

Other denominations have special services as well.  Typical Baptist Christmas services, for example, involve plays and singing by the kids in the Church.  It tends to draw family members that do not attend.

Christmas is the celebration of the Lord and Savior - the entire reason for the existence of the Church in the first place, so it kind of makes sense that there would be some special pomp and circumstance surrounding the celebration.  The same can be said for Easter.  

As for why people visit church on Christmas, that's far more personal and individual.  As such it's probably not on-topic to speculate, but reasons I've heard and my own reasons when I wasn't' a Christian included:


It helps you get into the Christmas spirit.
Tradition
From an atheist perspective: It's a nice thing to do to show support for that Christian relative of yours that's always bugging you to get saved and turn your life over to Jesus.  it's only an hour or so and it tends to seem to make them happy to have you there, so why not?
Maybe people already are in the Christmas spirit, and therefore are more willing to go.


I'm sure there are dozens of other reasons.  Again, that's all unique and individual, and hard to answer.