Simple Examples:
* The new software is compatible with all major operating systems.
* My new phone is compatible with my old headphones.
* They are a compatible couple, with shared interests and values.
More Complex Examples:
* The two companies merged because their business models were compatible.
* The doctor prescribed a medication compatible with the patient's allergies.
* The design of the furniture was compatible with the modern aesthetic of the house.
You can also use "compatible" with the preposition "with":
* This printer is compatible with Windows and Mac computers.
* Her personality is compatible with his.
The best way to use "compatible" depends on the context and the message you want to convey.