How do I make my object look smooth without using the subdivision surface modifier? Sorry if I phrase this incorrectly but I was following a tutorial for a character model (the link should be below, end of part 4 and beginning of part 5) and I don't understand how he got the body to look so smooth. I tried applying the subdivision surface modifier but it only made my object look flatter, like this:
It originally looks like this in sculpt mode: 
And like this in Object Mode:
I want it to look smooth and not so blocky like in the tutorial: 
https://www.youtube.com/watch?v=wdsY4BtlcpQ&amp;list=PLvgIVNDU-Dxjb3eukDF5W0l0-6ShO9OiM&amp;index=5

Thank you for your help

P.S.: Sorry if I added too many pictures to get my point across. Also, I am a novice at Blender, and yes, I know, even without the blockiness of my character's body, it doesn't look like the one in the tutorial. I'm doing my best so please disregard that if you can.
I think most of the effect you're seeing comes from switching the default Shade Flat (which uses a single Normal per face when shading, from what I can tell) to Shade Smooth (which interpolates -- that is, smoothly averages -- between the face normals to get useful normal information in between the basic face normals).

For example, here's an Icosphere with Shade Flat on by default:



And here's what happens with Shade Smooth turned on (in the sub-menu indicated in the above image):



Secondarily, you may also find it useful to use the Smooth Vertex (from the Vertex sub-menu in Edit Mode) transformation to improve the smoothness of the model. I believe Daniel Kreuter, the YouTuber who made the tutorial you're following (I recognize the mesh-work, having followed the same tutorial myself in the past), makes liberal use of Smooth Vertex; he explains this in one of the tutorial's earliest videos (and that he likes to hotkey it to the 'Q' key).