I am confronting myself with a problem that I do not know how to interpret. I am doing a project using Android and Google App Engine. When I am trying to save information in
This is not an answer, but is too long for a comment and may be helpful, so here goes...
First, it's hard to tell what's going on here because there is limited info. Try to post more of the logging either on the client or on the server. Go to the server console and get more info, for instance, or add debugging in the client. I'd start by trying to make the POST from a debug/testing tool like hurl.it: http://www.hurl.it/ (hurl is a nice web front end to a command line HTTP client named curl, see curl itself if you want a more advanced peek). That way you can test and poke around and make SURE your server side works as you expect with a generic client. Then build other clients (like Android).
Second, "Apache-HttpClient/UNAVAILABLE" is not an error, it's just the user-agent String -- so ignore that part. (UNAVAILABLE is where the version typically is, but some impls don't have access to the version sometimes, it seems.)
Third, the 405 response code IS an error, it means POST is not allowed at the server you are trying to POST to. That can either be because POST is not allowed at all on said server, or you're violating some security policy (such as same origin).
If it's AppEngine, first check the APP you are using and make sure it supports POST (look for info on how to do POST at AppEngine to solve, for instance: google app engine: 405 method not allowed). When you say you can do it directly from a "servlet," do you mean that a POST from a different client works? (Servlet is a server side technology, so that's a little confusing.) If so then make sure your Android app is doing the client part the same way to the same host (multipart vs urlencoded, etc).
For a little more on the security stuff, which could be involved, see this post: Google App Engine + jQuery Ajax = 405 Method Not Allowed.