Eternal Sunshine was directed by Michel Gondry, a French writer and director who got his start in the music video industry. Being the drummer for a rock group called Oui Oui, he made some animated music videos like this one, and after catching the eye of other artists, he started directing music videos for them, too. His clients include some big names like Radiohead and Daft Punk.
As you can tell from these videos, Gondry has a very particular (translation: really weird) artistic style, and much of this directorial aesthetic can be seen in Eternal Sunshine. We don't want to go into too much detail here, but Gondry's manipulation of perspective (especially seen in the Radiohead video) is used a lot in Joel's younger memories. Also in the Radiohead video, you can see Gondry's use of extensive panning to seamlessly transition the viewer's focus while keeping one seemingly continuous shot.
Eventually, Gondry made his way from music videos to directing features films. Now, Gondry's an indie director. His one foray into big-time Hollywood directing was with the superhero parody movie The Green Hornet, which he co-wrote with Seth Rogen.
When given full directorial control, Gondry has directed some well-received indie flicks like The Science of Sleep, as well as some documentary-esque films like Is the Man Who Is Tall Happy?, an illustrated conversation with famous linguist Noam Chomsky (yes, apparently if you're smart enough you can be a linguist and be famous). It turns out Gondry is also a talented animator. What can't this man do?