Talk about "possible worlds" is a metaphorical way of discussing ways-that-things-could-have-been so, e.g. to say that there is a possible world in which Bill Clinton never got into politics, or law, but spent his career as a shoe salesman in Little Rock is just to say that it might have been that Bill Clinton was a shoe salesman. 

There are lots of things that could be or could have been: the US could have invaded Iraq in 1991—we can’t change the past but things could have been different--Russia could revert to Communism, keep its current regime or, for that matter, establish a monarchy; you could have dropped out of high school, you could now be at the beach asleep rather than reading this handout. All these states of affairs, however likely or unlikely, are logically possible. Even some truly bizarre states of affairs that aren't even physically possible are logically possible: the moon could be made out of green cheese and things could go faster than the speed of light. 

Not everything that is logically possible, or even physically possible actually happens, of course: you didn't drop out of high school and you aren't now asleep on the beach. Continuing the metaphor of possible worlds, talk about the "actual world" is a way of indicating that we're describing the way things actually are. So, in this terminology, the actual world is one among many possible worlds. To say that a state of affairs obtains (holds true, is the case) at the actual world is just to say it's the way things actually are. Bill Clinton went to law school and was elected president; you went to college and are currently reading this handout. Whatever is actual is possible--but there are many things that are logically possible that are not actual. 

Given this broad definition of logical possibility you may ask: what, if anything, isn't logically possible? The traditional answer is that what is logically impossible is a 'contradiction in terms'--something that is 'inconceivable'. So it is logically impossible that there be a round square or a married bachelor. Similarly, given your instructor's scheme for curving grades on tests--where the high score, whatever that is, is 100%=A, it is logically impossible that there be a test in this class where no one gets an 'A'.  Using the possible worlds metaphor we would say that there is no possible world at which there are round squares or married bachelors, and no possible world in which the highest grade for a test (whatever it is) is an 'A' but no one gets an 'A'.  

Necessary truths are true at all possible worlds hence there are no possible worlds in which propositions that contradict them are true. Mathematical truths generally are held to be necessary truths so we should say "2+2=4" true at all possible worlds whereas "2+2=5" is not true at any possible world. It is logically impossible. 

Traditionally the test for determining whether a state of affair is logically possible is to try to conceive of a possible world in which it holds true--to imagine or describe such a world. 

 Necessary and Contingent Propositions

1. Contingent Propositions

To say that a state of affairs is ‘logically possible’ is to say that it is possible in the weakest, most minimal sense: it is to say that it is ‘conceivable’ or, alternatively, that it does not involve any ‘contradiction’. If something can be imagined, even though it may be physically impossible, it is logically possible.

'Contingent' just means 'not necessary'. Some propositions are contingently true: as a matter of fact they are true, but they could be false, i.e. it is conceivable or possible that they be false. Some contingently true propositions are:

(1) San Diego is in California.

(2) Barak Obama was President of the United States on January 1, 2016.

(3) The earth goes around the sun.

(4) On earth, things fall at 32 feet per second per second.

Logical possibility is broader than physical possibility: there are states of affairs that are logically possible but not physically possible. Laws of nature are only contingently true since it is conceivable that things be otherwise. It is not physically possible for anything to go faster than light, but it is logically possible.

The following propositions are contingently false:

(5) San Diego is in Texas.

(6) Vladimir Putin is President of the United States on January 1, 2016.

(7) The sun goes around the earth.

(8) There is no such thing as gravity: everything just floats around.

2. Necessary Propositions

To say that a proposition is necessarily true is

(9) Either San Diego is entirely in California or San Diego is not entirely in California.

(10) Que sera sera. [Whatever will be, will be.]

(11) All bachelors are unmarried.

(12) 2 + 2 = 4

To say that a proposition is necessarily true is to say that its denial (or "negation") is not even logically possible. The following propositions are necessarily true in this sense:

(13) San Diego is both entirely in California and not (entirely) in California.

(14) Some things that will happen will not happen.

(15) Some bachelors are married.

(16) 2 + 2 = 5

Notice, the denial ('negation') of any necessary truth is necessarily false and the denial of any proposition that is necessarily false is necessarily true.

A Puzzle About Logical Possibility: How can there be necessary truths? 

On this account, how can anything be logically impossible?!!? We can always describe a "world" in which a given state of affairs obtains, if we're clever. Take "all bachelors are unmarried": I can describe a world were "bachelor" means "male under 30" and such a world is one in which there are married bachelors, right? Similarly "2+2=4" and "2+2=5": it's just a matter of how you define the symbols, right? It's like your grading scheme: you could have set the curve differently so that no one got an "A." So, with logic, all things are possible...   What's wrong with this picture?