Smart Object feature was first introduced in Photoshop CS2. It basically allows you to  rotate, scale and transform raster and vector

graphics non-destructively. You can rotate,scale and transform raster or vector graphics in photoshop without smart objects but the key

here is non-destructively. Let's start by opening an image in Photoshop

Press Ctrl+T to resize, rotate an image and apply the transformation. Now if you would press CTRL+T again and resize the image it would lose it's quality like the image below

Now let's use smart object and see how smart object work in Photoshop.

To open an image as smart object, select File>open as smart object and select your image as shown below.

Your smart object image will have a little icon in layer panel to identify it as smart object.

Repeat the process of resizing the image by pressing Ctrl+T and apply transformation. Now you decided to resize the image again perhaps increase the width a bit but you know that doing so will reduce the quality of the image but as it's smart object you can resize it, rotate it or wrap it without losing any quality. This is called smart object.

A smart object resized again without any loss of quality.

Another cool thing about smart object is that you can replace the smart object with another image with the same dimensions as the original image. To do so Go to Layer>Smart Objects > Replace Contents and select your image.

Difference between Smart object and normal image after free transformation.