I have designed a logo in Adobe CS5.1 Illustrator using the ellipse tool to create shapes, filled with colour and then added my type.
The file is saved as
Is it automatically a vector image?
There's no way to know for certain without seeing the actual file.
As a guess, I'd say yes, most likely.
Just because you used Illustrator does not automatically mean everything is vector. If you used, blurs, feathers, glows, drop shadows, placed images, photoshop effects within Illustrator, etc. - none of that is vector.
If you haven't used any special effects, you should be good, but as Scott said, all special effects will Flatten to Raster (pixels).
The Flattening settings will also effect what gets Rasterized on export.
If you're not sure, use the Flattening Preview.
You can manually Flatten an object be selecting it and going to Object / Flatten Transparency
You can also switch to Outline View (Command + Y).
Anything you can see in Outline View will be Vector. Any object that is just a box, or not displayed (like Effects) is not Vector.