Why do I get a "permission denied" error while installing a gem?

Your Ruby is installed in /usr/local/Cellar/ruby/....

That is a restricted path and can only be written to when you use elevated privileges, either by running as root or by using sudo. I won't recommend you run things as root since you don't understand how paths and permissions work. You can use sudo gem install jekyll, which will temporarily elevate your permissions, giving your command the rights needed to write to that directory.

However, I'd recommend you give serious thought into NOT doing that, and instead use your RVM to install Ruby into your own home directory, where you'll automatically be able to install Rubies and gems without permission issues. See the directions for installing into a local RVM sandbox in "Single-User installations".

Because you have RVM in your ~/.bash_profile, but it doesn't show up in your Gem environment listing, I suspect you either haven't followed the directions for installing RVM correctly, or you haven't used the all-important command:

rvm use 2.0.0 --default

to configure a default Ruby.

For most users, the "Single-User installation" is the way to go. If you have to use sudo with that configuration you've done something wrong.

I wanted to share the steps that I followed that fixed this issue for me in the hopes that it can help someone else (and also as a reminder for me in case something like this happens again)

The issues I'd been having (which were the same as OP's) may have to do with using homebrew to install Ruby.

To fix this, first I updated homebrew:

brew update && brew upgrade
brew doctor

(If brew doctor comes up with any issues, fix them first.) Then I uninstalled ruby

brew uninstall ruby

If rbenv is NOT installed at this point, then

brew install rbenv
brew install ruby-build
echo 'export RBENV_ROOT=/usr/local/var/rbenv' >> ~/.bash_profile
echo 'if which rbenv > /dev/null; then eval "$(rbenv init -)"; fi' >> ~/.bash_profile

Then I used rbenv to install ruby. First, find the desired version:

rbenv install -l

Install that version (e.g. 2.2.2)

rbenv install 2.2.2

Then set the global version to the desired ruby version:

rbenv global 2.2.2

At this point you should see the desired version set for the following commands:

rbenv versions

and

ruby --version

Now you should be able to install bundler:

gem install bundler

And once in the desired project folder, you can install all the required gems:

bundle
bundle install