adobe photoshop - How can I make my logo look crisp on the web?

I'm launching a company and my logo looks really pixelated and low-quality compared to some others. For example, http://talentegg.ca/ has a really crisp logo, how can I get mine like this?

Mine, which is better? Both of which can be improved though

1. 
   

   

   
   


2. 
   

   

   
   


3. 
   
   

   Illustrator version -

   
   


4. 
   

   saved for web version -

   
   

Please help!! I have Adobe Photoshop CS6