When assessing the overall health of a tree there are 2 categories to take into consideration: circulatory and structural. It's possible for a tree to be healthy in one category, yet unhealthy in another.
Circulatory health is how efficiently a tree is transporting water and nutrients. This occurs in the wood just beneath the bark. It is also highly dependent on root health and development.
Structural health is a measure of how stable the tree is. This is mainly affected by the growth pattern (lean or one-sided growth) and the condition of the heartwood (See "A Heart of...Wood?").