To add a few comments to Keith's valuable points:
- You usually want power supply nets wired with a larger trace width than signal nets. For auto router operation, it's required to set all design rules, including net properties before you start the routing, it's also highly recommended with manual routing.
- The capacitors C1 - C4 are apparently intended as bypass capacitors. To be effective, they should be placed near the components that need supply bypassing.
- I guess, the Eagle autorouter stopped due to an unsuitable routing grid and/or minimum distance setting. It should be able to route most nets of the present design.
But I agree, that manual routing gives neater and mostly better results. A compromise may be to route power supply and critical analog/high speed nets (if any) manually, protect them and let the ordinary signal nets to the autorouter. I'm not sure about the results with Eagle, but it's a promising method to speed up PCB design with industry standard tools like Specctra.