So you have a tree in question, what do you check?
First, The Whole Tree. If the tree has been leaning for a long time, it may be fine but look for other signs of decay or danger. If a tree is leaning considerably more than it has in previous months, weeks, or days, seek a professional immediately.
Next, The Ground. A major concern right now is oversaturated earth and it eroding away, creating an unstable root system and causing a tree to fall. Look where the trunk meets the ground. If you see cracked, raised, or an exposed roots, it may be in the process of uprooting.
Then, The Trunk. There are many signs of a damaged, diseased, or decayed trunk. Splits/cracks, missing bark, shallow to deep cavities, sawdusts from ants or beetles, fungus or mushrooms, are dangerous if on the trunk but can also be present anywhere on a damaged tree.
Finally, The Canopy. Missing bark, early changing leaves/pine needles and limbs falling are typically a tale-tell sign of something wrong with a tree, especially if it has failed your other observations. If you feel you have a troublesome tree, contact a professional and let that stress disappear.
Reminder: This guide is to help assess if you need to consult a professional. Always let a professional decide whether a tree is safe or not. If you have any concerns about a tree, it is better to be safe than sorry!