While Tchaikovsy's music lacks the depth and harmonic complexity of other Romantic composers, he is the master of melody. No one could spin a tune like he could! He also possessed a keen sense of drama that makes his music both fun for musicians to play, and extremely popular among audiences. Almost everything he writes is exciting, with fast and rhythmic passages that contrast with wonderfully lyrical moments. "Romeo and Juliet" is a great example of his style.
His ballet music is timeless--every child should see "The Nutcracker" more than once, and he immortalized the role of the dying swan in "Swan Lake." His music is so entrenched in our culture--you hear it in cartoons, in commercials, in movies, etc.--that people often don't recognize it as his. Your kids have probably been exposed to Tchaikovsky numerous times already without you even trying!
A funny anecdote to illustrate: When I was in high school, and the Magic Valley Symphony performed the Suite from "Sleeping Beauty," my friend exclaimed, "I had no idea that Tchaikovsky took his theme from the Disney movie!"
In spite of, or because of, his popularity, Tchaikovsky was considered a sell-out by other Russian composers. While Tchaikovsky was working on "Western" forms, like symphonies and concertos, other Russian composers were promoting Russian music, with Russian melodies, Russian harmonies, and Russian forms--avoiding the structures and conventions used by European composers. There was a group of composers called "The Russian Five" or "The Mighty Five" who represented this movement, and pitted themselves against Tchaikovsky--the "Westernizer". The group was made up of Mily Balakirev, César Cui, Modest Mussorgsky, Nikolai Rimsky-Korsakov and Alexander Borodin. You might be interested in exploring some of their works. For quick reference, think of "Night on Bald Mountain," the famous work by Mussorgsky that was used in "Fantasia"; and the popular "Scheherazade" by Rimsky-Korsakov (probably the most choreographed piece in Ice Skating history).