“An algorithm must be seen to be believed”, these are the words from Donald Knuth himself, the master who has written the best and comprehensive books on Algorithms. Here are some resources which would help you in visualising some common algorithms:
Algorithm Visualizer: The visualisations are pretty good on this one. You can see the control step through each statement on the javascript code block and on another side the input data changing places/shapes. Very good tool that provides cool controls like pause/play/rewind.

VisuAlgo: VisuAlgo was conceptualised in 2011 by Dr Steven Halim as a tool to help his students better understand data structures and algorithms, by allowing them to learn the basics on their own and at their own pace. VisuAlgo contains many advanced algorithms that are discussed in Dr Steven Halim’s book (‘Competitive Programming’, co-authored with his brother Dr Felix Halim) and beyond. Today, some of these advanced algorithms visualization/animation can only be found in VisuAlgo.

Algomation: Algomation is a platform for viewing, creating and sharing any type of algorithm. All algorithms on on the site are public and can be viewed and shared by any user of the site. Registered users can create new algorithms or fork existing one.

Algorithm Wiki: This wiki is an experiment in making algorithms interactive on the web. The algorithms on this wiki are not just some clever animations but are built on a real Javascript interpreter, and the visualisations are based on the actual Javascript code running. The code is guaranteed to work.

Vamonos: A dynamic algorithm visualization in the browser. It is also a library for generating browser-based visualizations of algorithms & data structures.

Data Structure Visualizations: The site has interactive animations for a variety of data structures and algorithms. The visualization tool is written in javascript using the HTML5 canvas element, and run in just about any modern browser — including iOS devices like the iPhone and iPad, and even the web browser in the Kindle! (The frame rate is low enough in the Kindle that the visualizations aren’t terribly useful, but the tree-based visualizations — BSTs and AVL Trees — seem to work well enough)

Some interesting videos containing the fusion of folk dance/music with Sorting Algorithms:

“An algorithm must be seen to be believed..”
-Donald Knuth