What did combatants on each side of the Civil War believe they were fighting for?

There are many
answers to your question.  However, the simplified view is that the south was fighting to
protect their economic livelihood which at the time was slavery.  The industrialized north was
fighting against slavery which they saw as an inhumane institution.

With that
being said, you could begin by looking at the governmental clash with the issue of slavery.  In
the Constitution, you have in Article II listed that slaves were counted as 3/5 of a person for
representation purposes only in the House of Representatives.  You also have the various
compromises authored by Henry Clay, such as the Missouri Compromise and the Compromise of 1850.
These compromises were about keeping the status quo and not tipping the balance of slave states
and free states. Even in Kansas, there was an outbreak of violence over slavery, which many
historians call Bleeding Kansas. It was an attempt to stack the deck in voting for or against
slavery and the settlers, both from the north and south,  to the region fought over their
beliefs about the issue of slavery.  John C. Calhoun with the Nullification Crisis tried to set
the south up to be able to over turn or nullify federal law.

The
industrialized north was concerned with the mistreatment of slaves. However, the north really
did not treat its workers much better than the southern slave owners treated their slaves. The
industrialized north had poor housing, low wages, and long hours for their workers; the only
difference was the north did not beat their workers. For the north was about protecting the
Union and stopping the mistreatment of the slaves.

A great document to look
at is the South Carolina Secession Statement.  Which you can find at the civilwar.org website.
 Here is what Calhoun wrote about South Carolina and he used the words of the Declaration of
Independence against the Union.  He stated, "...A struggle for the right of self-government
ensued, which resulted, on the 4th of July, 1776, in a Declaration, by the Colonies, 'that they
are, and of right ought to be, FREE AND INDEPENDENT STATES; and that, as free and independent
States, they have full power to levy war, conclude peace, contract alliances, establish
commerce, and to do all other acts and things which independent States may of right do.'"
Calhoun believed that war was the only way to protect the southern way of life -
slavery.