What is an FAA Notice? Does it have legal force?
As SSumner pointed out, the FAA produces all kinds of paper - some of which is "regulatory" and/or have legal force, and some of which isn't.  I won't go into all of the specific document categories, but I'll talk about a few that most of us will find interesting.


Handbooks and Manuals
The FAA actually publishes full-length books, if you know where to look!
The FAA has written books on flying an airplane,  flying balloons, risk management, buying &amp; owning an airplane, and a whole bunch of other topics which provide broad information that you can use to familiarize yourself with a topic.
Advisory Circulars
Advisory Circulars (ACs) are the FAA's way of explaining their regulations.
They're broken down into sections which generally match up with the FARs they're interpreting.  Some ACs may rival the lengths of handbooks, but they are much more narrowly focused, generally dealing with a single topic and often much easier to understand than the legislation they're interpreting.
Airworthiness Directives
When an unsafe condition is identified in an aircraft or component which requires action the FAA can issue an Airworthiness Directive (AD), requiring that specific actions be taken.
Often the required action is compliance with a manufacturer's Service Bulletin (or compliance with a service bulletin is also an acceptable method of compliance with an AD).
There are also Special Airworthiness Information Bulletins (SAIBs), which are like "Mini-ADs": The FAA is calling operators' and mechanics' attention to something which may not merit an AD. SAIBs are informational - you don't have to comply with them - but it's certainly a good idea to read them and pay attention to the things they're highlighting. 
Orders
An "Order" describes procedures used by the FAA. For example, JO 7110.65U describes various aspects of how ATC works. 
Procedures described in Orders may be modified by Notices.
Notices
A "Notice" is just an agency's way of saying "Yo! Pay attention! Something's happening!"
Your question seems generally concerned with internal notices which I linked to in the title. These generally modify Orders, like this one.
The FAA can also issue "Public Notices" about changes that affect airmen and the   general public, or pretty much any other topic they'd like.  Public Notices are published on faa.gov as well, but the easiest place to find those is through the FAA Safety Team's SPANS system which is searchable and has a much nicer interface than digging through the FAA site by region.




As for whether the non-legislative documents have legal force the answer is a very well-qualified "maybe".  Some certainly have the force of regulation – Airworthiness Directives being the most obvious – and for our purposes I think that's what you're really concerned with, because the law says you have to comply with the regulations.
Orders and Notices are similarly "regulatory" for the agency: The FAA has to comply with its own orders, so ATC has to be operated the way JO 7110 says it's to be operated.

For stuff like ACs, Handbooks, etc. we discussed this in chat a while back talking about the answer to FAA Order 8900 and the FSMS - the gist of it is that the non-legislative documents the FAA produces are generally interpretations and guidance, but they are "effectively regulatory" in that if you go against them you need to have an excellent reason for doing so, and be prepared to back it up with appropriate justification and evidence if asked. 

As a classic example, alterations to aircraft are expected to be performed in accordance with manufacturer's maintenance procedures, or "other methods, techniques, or practices acceptable to the Administrator" which are described in AC 43.13 (43.13-2B deals with light, non-pressurized aircraft and is an interesting read).

A mechanic doesn't have to comply with the guidance in AC 43.13, but if a mechanic were to modify an aircraft without conforming to that guidance they would need to supply supporting documentation as to why their work is acceptable, such as a statement from a Designated Engineering Representative, otherwise the FAA could deem the work "unacceptable" (and from a liability standpoint, the mechanic could wind up in serious trouble should something go wrong with the aircraft as a result of that work).