The term "pixelated" is a bit ambiguous. Basically it means the same as "rasterized", but we often use "pixelated" about images where the pixels are particularly visible. For example when there is a lack of anti-aliasing or when the image has been scaled up so each pixel in the image is drawn using several screen pixels resulting in a chunky appearance.
Everything you see on a screen is displayed using pixels. So vector graphics and type have to be rasterized to be able to be displayed.
When you are working in a vector application like Illustrator, the data of your vector graphics is of course stored as points in a coordinate system so you can scale and manipulate objects freely without loosing quality. But in order to display your graphics on screen they have to be constantly rasterized as you change zoom level and pan around.
Saving or exporting as .ai, .eps or .pdf and other vector formats preserves the vector qualities of your artwork, but when you export a raster image, you decide on an image size and the graphics will be rasterized to that particular resolution for good.
A vector file can have a relatively small file size, but still be very demanding for the computer to render on screen. A raster file will in many cases be larger, but because it has already been rasterized, it's often quicker to display.
In a raster editor like Photoshop you can draw vector objects or place Illustrator files which can be scaled up and down without a loss in quality of the vector shapes themselves. But a Photoshop document always have fixed pixel dimensions so the vector graphics will be rasterized at that particular resolution! In other words, they will only look sharp when you view the image at 100%. If you zoom in, you will see chunky pixels.
I don't know much about video editing software like After Effects, but it must be possible to use scalable vector shapes there as well. But only in the editing phase. In the end you will export a video in a certain resolution, so the vector will end up as pixels. A video which can be scaled up endlessly and keep looking sharp doesn't exist.