link_to is used in your view, and generates html code for a link
<%= link_to "Google", "http://google.com" %>
This will generate in your view the following html
<a href="http://google.com">Google</a>
redirect_to and render are used in your controller to reply to a request.
redirect_to will simply redirect the request to a new URL, if in your controller you add
redirect_to "http://google.com"
anyone accessing your page will effectively be redirected to Google
render can be used in many ways, but it's mainly used to render your html views.
render "article/show"
This will render the view "app/views/article/show.html.erb"
The following link will explain the redirect_to and the render methods more in detail
http://guides.rubyonrails.org/layouts_and_rendering.html
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