Picnik is shutting down in April and people are panicking.
Picnik is/was fun to play with, but on a very amateur level, so it certainly isn't irreplaceable and it is already incorporated into Google+, anyway.
Some people specify they want only a website (and always free !), rather than including resident programs, too.
If you are one of them, then my question is: "What is your issue about installing a program?"
Do you realize that with websites like picnik etc you have to actually first UPLOAD each and every photo you work on and then DOWNLOAD each and every photo again?
That is a lot more up/downloading in the long run than with any resident program.
When your internet is down, you have nothing.
The programs that live on your computer such as Gimp (free), Paint.net (free), Photoshop Elements or Photoshop proper - currently CS5 - are much more powerful and useful. For starters, you could even try out Photoscape (also free).
If you think about it for a moment, you should understand which one is your better option by far. However, if you are determined to use only a website, then you can try these for starters:
http://www.picture2life.com/ or http://fotoflexer.com/ or http://www.sumopaint.com/ or http://aviary.com/ or http://pixlr.com/
The last three are your best bet for more complicated tasks.
P.S. questions about such programs or websites get asked countless times every single day on here.
If you type "editing websites" into the search box, you'll get literally thousands of answers right away. http://answers.yahoo.com/search/search_result;_ylt=AphDJyOJg48kEtAvFZfmoC2NxQt.;_ylv=3?p=editing+websites&submit-go=Search+Y%21+Answers