The issues the Synod of Dort dealt with in 1618/19 were already listed by Elle. The Reformed response to them can be summarized as "the Five Points of Calvinism". English-speaking Reformeds (and Presbyterians, for that matter) usually remember them from the acronym TULIP:
Total DepravityWe believe that man by nature, apart from the grace of God, is dead in sin, and in no way able to make up for his sin and unfaithfulness against God. In short, we are miserable sinners, even our will is enslaved to sin, and our best works fall short of the just standards of our holy God.
Unconditional ElectionGod has given Jesus Christ as the atonment for sin; all who believe in Him are saved. Now ultimately, this saving faith does not come from man himself. Indeed, when "natural" man is presented the gospel, he will reject it and despise the sacrifice of God's only Son. Faith itself is a gift from God; the willingness to believe is due to the work of regeneration, by which God changes our heart and turns our will his direction.
God does not regenerate all men, not even all people who are raised in a Christian family. He chooses whom he saves. This is the profound Biblical truth of "election". The Arminians argued that God's election is really based on his advance knowledge about who will want to believe and who will not. The Reformed, however, hold fast to the truth that God's election is not based on anything in us, but only on his good pleasure. There are no conditions that we must fulfill to be elected.
Limited AtonementThis is the teaching that the atoning work of Christ extends to the elect (= those who believe or will believe) only. Only the people chosen by God partake in the salvation. This is completely according to God's plan and purpose.
Irresistible GraceThe doctrine of irresistible grace is probably the most disputed issue between Reformed believers and virtually all others. We believe that God's gracious work of regeneration (which he only works in the elect) is not dependent of man or the choice of man. When God comes to man and works in his heart, he will not ask permission (after all, we would say "no"). God's grace is all-powerful, and when he touches our hearts we will certainly turn to Him and believe Him. (The Arminians taught, that somehow man must "invite" or "accept" the regenerating Spirit of God in their hearts. This thought is also common among evangelicals. It presents, however, a far too positive view of the capabilities of man.)
Preservation (or Perseverance) of the SaintsThis Biblical doctrine is the conforting truth that God, who began a good work in us (= his elect), will also complete it. When God has chosen for someone, nothing can separate that person from his love. God can and will provide everything to keep the true believer close to him. The concrete message is, that we need not fear to fall away from God's grace; he does not tempt us beyond our capacities, he will give strength and grace where necessary. Even if we fall in sin, God is faithful.
Many more can be said about these issues. Different Reformed denominations often disagree about the fine details of these teachings. In some churches, the rich content of this Calvinistic point of view have been watered down; others make a charicature of it ("Hyper-Calvinism"). But these are the mainlines. If you want to know the historical context, try to find a copy of the Canons of Dort. There are several online versions; try, for example,
www.prca.com or
www.crc.com.