That said here is my best interpretation of it all. It should help put it into some kind of context at least.
A general, unexpected change, or multiple changes in your search result position. Can happen daily, can last for weeks, can have very severe swings . For more mature sites it is assumed to be Google making changes to the algorithm that may not always be successful. Further, multiple minor adjustment occur as Google get it right.
For new sites it is likely that the Algorithm is very different to mature sites. Initially google will only have on-page SEO to measure and it seems to give an initial boost to new sites to give them exposure and be in a position to start getting natural links. As the site gets over few weeks old then the Algo changes and off-page SEO becomes more important. During this period a web site's search position can be very volatile.
This is one that seems to have the least information and the most confusion. From what I have read and from my own experience it seems that “fairly new” sites can get “sandboxed”. This is NOT de-indexed but may be very low or even nowhere in the search results. You may see some long tail/related keywords ranking but generally nothing. This may last 3 months or so.
My best guess is that Google sees something it is not happy about with the site i.e. unnatural backlinking and so puts it “on-hold”. That is likely to be some overzealous link building where the issue is transitory and so a “time -out” will naturally resolve the issue
As this period is quite long it can be difficult to know if your site is sandboxed or penalized (see Algorithm Change below). So it may be worth treating it as penalized.
The results of being penalized will seem to be exactly as sandboxed but you have a “penalty” for misbehaving that will stick. Maybe linked to some very bad sites, massive abuse of backlinking “rules” etc. If you know you've been a bad boy then fix the issues and apply for re-instatement as in de-indexed (What To Do?)
A brand new site will of course not be indexed but by adding a few links, a sitemap and few social bookmarks you should get your site indexed in a few days. People saying they cannot get indexed in weeks or months are either doing something VERY wrong or have actually been de-indexed.
If you were indexed and then find you are no-longer indexed then you have been de-indexed.
Type “site:yoursite.com” into Google search. If you get no results then you have been de-indexed.
This will only happen if you have done something really bad in Google's eyes. Which could be almost anything.
There is one final option and that is that you site could be “dropped” in the search results due to a “correct“ algorithm change. This would be a permanent change. I.e the Recent Panda/Farmer updates saw many article sites and derivatives drop dramatically in the search results.
What To Do?
If it's Google dance or sandbox then it's general just a matter of being patient. It may be worth still adding a steady flow (not blasting) of backlinks to the site during this period and you are likely to see higher results than before once you come out the other side.
If It's a permanent change in results then you will need to look at what the new algorithm is targeting and change your site to better fit the new “rules”.
If you have been de-indexed all may not be lost. If your site was earning good money then it will be worth “fixing” your issues and asking google to re-instate you in their good grace. There is an application form for this, and instructions you can find in webmaster tools. Google seem to have a “confession is good for your soul” approach and expect you to come clean about any indiscretions. In fact it may be advantageous to do so as it shows you have fixed things and are now “good guy”.
On the other hand if it was not making much many anyway you may be best to simply scrap the website and get rid of the domain name (wash your hands of the whole thing). That way at least you won't be associated with a “bad” site as that may actually effect your other or future sites.
If you haven't already done so then sign up and register your sites with Google Webmaster tools. That way you may even get some information on problems with your website. For instance if your site gets infected with Malware then Google will certainly penalise or de-index your site. If you are register with Webmaster tools you will get notification and instructions on how to fix it. See
Cleaning your site - Webmaster Tools Help
If you want to ask Google to review your website and get any penalties or de-indexing lifted then use the following form
It's pretty much a freehand form and the best way to complete it is to list all the quality guidelines and the points mentioned at the top of the form
Under each heading explain how you have reviewed for compliance or found non-compliance and how you fixed it.
You will not get a reply, you will not get confirmation of re-instatement, you will not get a rejection notice and you will not get any help or discussion. So get the form right first time and cover everything. If successful you will see your site re-appear or improve in the search results.