Can I download Jekyll CMS, like downloading WordPress ?
I’d like to try to install it on my local server
It is much better to use the recommended ways to run Jekyll. Recommend way to run Jekyll saves you from myriad of troubles and consistently generates the desired output files.
If your use of Jekyll does not fit any recommended way - you might want to share it, so the community would have a chance to give you their best advice.
Jekyll isn’t a CMS like WordPress. It’s a program that runs on your computer and outputs static HTML files from text stored in Markdown files, which you then upload to your server.
To install Jekyll on your computer, follow the installation guide.
I’m trying to start a website from my Android phone. So no command prompt
In that case you might be more interested in GitHub Pages. It uses Jekyll internally to build the website in your GitHub repository whenever you push changes to it.
Wait, are you trying to run Jekyll on your Android phone? 
You can jailbreak your phone and install a terminal program, but you’d also have to install Ruby. I don’t know if there’s an Android port.
Chromatical’s suggestion is good, use Github Pages and let it run Jekyll for you. Assuming, that is, Android has a Git port.