In what ways did the Battle of Trenton transform attitudes toward the War of Independence on both sides of the Atlantic?

Basically, the victory that Washington won at Trenton
inspired people both in America and Europe to have greater confidence in Washington
himself and in the potential for the rebellion to succeed.  This had some important
consequences.

In the colonies, people had been losing
confidence in Washington.  Many soldiers were going to be leaving the army after their
enlistment terms were up at the end of the year.  The victory inspired many to stay and
convinced others that the cause was not lost.

A similar
thing happened in Europe.  There, the most important consequence was that the French
started to take a more active role in supporting the Patriot cause because they believed
that there was a serious hope of victory.

Please follow the
link -- it has a good discussion of these changes.