Question: I know that it is required to first create a database of references stored in a `.bib` file. Mendeley exports my reference list in `.bib` format. But how can I know what to write in curly brackets in `\cite{}` to refer to a particular author? Do I need to keep the `.bib` file open and find the relevant name to cite? Or am I missing something? What is your recommendation for referencing in LaTeX?

Answer: 
Another simple way is to select the file in Mendeley list, then right-click and select "copy as", then "LaTex citation command" (which corresponds to ctrl+K), and last paste the thing inside the brackets of \cite{}
The other answers work but I've found it much easier to use the LaTeXing plugin for Sublime text editor. From within Sublime you can search the titles and authors of all the papers in your online Mendeley account whenever you use \cite{}. 

<http://docs.latexing.com/stable/tutorials/setup-mendeley-with-latexing.html>

It automatically creates a .bib file containing just the papers you have added using Sublime. Unlimited (so far) free trials are available for both Sublime and LaTeXing.