When something is stable, it's fixed and steady. If you needed advice, you'd probably go to your most stable friend, the one least likely to act crazy or be easily upset.
Whether you're talking about an object or a person, the adjective stable implies reliability and strength. You can describe a government as stable, or a relationship, or a desk. A completely different meaning of stable is the noun "building used for housing horses or other animals." Both senses of the word come from the Latin stabilis, "firm or steadfast."