In the West "anime" means "animation from Japan." Myanimelist, a Western anime and manga databasing site, refuses to list a series unless it's Japanese (to the chagrin of Avatar: the Last Airbender fans).
However, the average Japanese person considers all forms of animation to be "anime," regardless of style or country of origin. I've even heard live action series like Kamen Rider, Metal Heroes and Super Sentai referred to as "anime" in conversation.
But industry insiders and purists (Hayao Miyazaki for example) define "anime" as a "limited-animation" style popularized by Japanese studios. Envisioned by Tezuka Osamu, limited-animation techniques lowered production costs while speeding up the production process, making animation feasible for television broadcast.