Dear Clement:

In the simplest terms Catholics worship only God. We venerate the Saints which means that we give honor to them.

In the United States, for example, we venerate (honor) George Washington and Abraham Lincoln because they were great Presidents, but we do not worship them.

Similarly, we venerate (honor) the Saints because they were great examples and role models of the Christian life.

That is the easiest way to explain this, especially to a person with an anti-catholic bent (well even to a well-disposed non-Catholic, and, let's face it, even to many Catholics).

This sort of explanation is probably best with those who are determined to believe the worst about the Church. Such people will only think that a detailed and technical explanation is rationalizing.

The problem, however, is if a person reads an older book they may see the "worship" language in reference to the Saints. Even though the older books refer to the worship of Saints, it is not referring to the kind of worship that is due only to God.

Even in the Bible, there are references to the "worship" of human beings (e.g., Genesis 37:7–9, Genesis 49:2-27, Exodus 18:7). Modern translations of these verses may or may not use the word "worship", but the original Hebrew does. Again, the use of the word "worship" in the Hebrew did not refer solely to the worship due only to God.

Both the Biblical languages and the English use the term "worship" in the broader sense. Especially in modern times, particularly among fundamentalist and evangelicals, the term "worship" is used in the restricted sense as an act due only to God.

The Church, in her ineffable wisdom sought to resolve this confusion. Thus, two terms: "Worship" (that is due only to God), and "Veneration" (that is giving honor to Saints).

For those who are open to the more technical explanation, here goes:

To quote from an article, Saint Worship, from Catholic Answers:

As the terminology of Christian theology developed, the Greek term latria came to be used to refer to the honor that is due to God alone, and the term dulia came to refer to the honor that is due to human beings, especially those who lived and died in God’s friendship—in other words, the saints. Scripture indicates that honor is due to these individuals (Matt. 10:41b). A special term was coined to refer to the special honor given to the Virgin Mary, who bore Jesus—God in the flesh—in her womb. This term, hyperdulia (huper [more than]+ dulia = "beyond dulia"), indicates that the honor due to her as Christ’s own Mother is more than the dulia given to other saints. It is greater in degree, but still of the same kind. However, since Mary is a finite creature, the honor she is due is fundamentally different in kind from the latria owed to the infinite Creator.

Oops, I forgot something when I first published this answer. Here is the additional info.

Concerning worship of objects, to quote an article, Do Catholic Worship Statues?, from Catholic Answers:

Catholics use statues, paintings, and other artistic devices to recall the person or thing depicted. Just as it helps to remember one’s mother by looking at her photograph, so it helps to recall the example of the saints by looking at pictures of them. Catholics also use statues as teaching tools. In the early Church they were especially useful for the instruction of the illiterate. Many Protestants have pictures of Jesus and other Bible pictures in Sunday school for teaching children. Catholics also use statues to commemorate certain people and events, much as Protestant churches have three-dimensional nativity scenes at Christmas.

I hope that helps.

God Bless,
Bro. Ignatius Mary