A non-standard company, is one that specializes in writing insurance that most companies don't want to write. There's no fixed set of criteria, and it's highly subjective.
The way to find out if your company is nonstandard, is to ask your agent.
As an example, Foremost Insurance writes a modified homeowners policy on low value homes, rental units, rowhouses, low credit score customers, and will even write homeowners with pitt bulls (with a bite exclusion). They are a non-standard carrier. They're admitted in every state, I believe (although I don't think they write new business in Florida), and are owned by Zurich, the third? largest insurance company in the world.
Foremost is NOT an excess (non-admitted) carrier. They ARE a non-standard carrier - they write risks that most insurance companies won't.