I wrote this pretty quickly and I'll come back to clarify and fix things – by all means ask questions and tell me what you think should be improved/added/removed etc.

1. Main clauses (also known as independent clauses)

In main clauses (that are not questions or exclamations), the verb goes in the second place in Swedish. This is called the V2 rule.

The 'places' in the sentence are sentence components, not necessarily just one word. It can be an adverbial or even a whole subclause. If you're in doubt, check whether you could get the same structure while exchanging the component for just one word.

Mormor och morfar äter 'Grandma and grandpa are eating'
– you'd have the same structure if you just said De äter 'They are eating'.

När jag kommer hem ska jag äta 'When I get home I will eat'
– you'd have the same structure if you just said Då ska jag äta 'Then I will eat' or something like that. [När jag kommer hem] is a subclause.

1.1 Adverbials in main clauses

There are three main places for adverbials in a main clause:

1. You can have the adverbial first. A typical example of this would be
   I morgon ska jag arbeta 'Tomorrow, I will work'
   Here, we start out from tomorrow – the idea is that I want to tell you what is going to happen tomorrow.
2. Time adverbials like i morgon also typically go at the very end of the sentence.
   Jag ska arbeta i morgon 'I will work tomorrow'.
   Here, the idea is that I'm telling you that I will work, and when it will happen (tomorrow).
   These two positions are typical for time and place adverbials and similar.
3. Adverbials that affect the sentence as a whole normally go in 4th place (they can also go first). We call those satsadverbial, 'sentence adverbials'. Typical examples of those are all kinds of adverbials that express the speaker's attitude towards what is said: lyckligtvis 'fortunately', tyvärr 'unfortunately', naturligtvis 'of course'. The adverbial inte, 'not', is pretty much tied to the 4th place.

We'll then have a sentence that starts like this:

1. first place
2. verb
3. subject (this place is empty if the subject is in the first place)
4. sentence adverbial

Examples:
Jag kan inte tyska 'I don't know German'
Jag kan faktiskt tyska 'I actually know German'
Ibland ville han inte gå upp 'Sometimes he did not want to get up'

This is pretty complex, but in short: generally, inte goes in this 4th place, but time and place adverbials such as på kvällen 'in the evening', i Spanien 'in Spain' go either first or last. Adverbials that express the attitude of the speaker can go in either of the 3 places.

1.2 Pronouns vs nouns as objects in negated clauses

In a positive sentence, there is no difference:
Jag älskar Björn or Jag älskar honom 'I love Björn' or 'I love him' – same order.

However in a negative clause, if you have a pronoun, two word orders are possible.
For a noun:
Jag älskar inte Björn 'I do not love Björn' (you cannot switch 'Björn' and 'inte')
For a pronoun:
Jag älskar honom inte is the normal word order but
Jag älskar inte honom is also possible. This word order is more special. It makes it sound like you probably love someone else instead. Too bad for Björn!

Also note: this does not mean that both word orders work in every sentence. There can be many factors, I'll just take one example: sentences with the construction inte A utan B 'not A but [rather] B'. E.g. Jag älskar inte honom, utan dig. 'I don't love him, I love you'. In this construction, you can't change the order between inte and honom because the construction inte A utan B is there to create a contrast between A and B, which won't work if we change the order between the words.

1.3 Particle verbs and objects

A particle verb and its particle are fairly easily separated by inte, but not as easily by objects. So in general, you'll see this:
Jag gillar inte måndagar 'I don't like Mondays' – easy, because it's a simple verb
Jag tycker inte om måndagar 'I don't like Mondays' – inte sneaks in between the verb and its particle, but the object goes after everything.
Degree adverbials can also sneak in between the verb and the particle, although they would otherwise have to be at the end of the sentence.
So we get:

Jag gillar måndagar mycket 'I like Mondays a lot' (impossible to put mycket right after the verb)
but
Jag tycker mycket om måndagar 'I like Mondays a lot'

1.4 Exceptions?

There are really very few exceptions from the V2 rule, but one that appears in the course is the word kanske, which can appear in second place before the verb:
Han kanske bor där 'Maybe he lives there'
The reason for this is probably that kanske is historically made up of two verbs, kan ske ('may happen').
If you really love exceptions, we found another apparent exception here, which I don't think really is an exception however. The V2 rule is actually very consistent.

1.5 Read more

There's much, much more to say about word order, even in the main clause. I think the diagrams on this site are very helpful: http://www.student.umu.se/under-studietiden/studieverkstad/skriftliga-uppgifter/skrivrad-och-sprakhjalp/ordfoljd-i-huvudsatser-och-bisatser/ Unfortunately they're all in Swedish, but as you learn, you'll get to understand them anyway, I think.

2. Direct questions

A direct question can either be answered with 'Yes' or 'No', or it contains a question word, such as var 'where', när 'when' hur 'how'.
In Swedish, questions are created by putting the verb before the subject.
Hon läser 'She reads'
Läser hon? 'Does she read?'
If there is a question word, it goes first of all:
Vad läser hon? 'What is she reading?'
A phrase can function the same way as a question word:
Hur många böcker läser hon? 'How many books does she read?'
Here, the phrase 'hur många böcker' functions the same way a question word would, so it also goes before the verb.

Indirect questions behave like subclauses, see below:

3. Subclauses (also known as dependent clauses)

In subclauses, the subject goes before the verb.

We already had an example here:
När jag kommer hem ska jag äta 'When I get home I will eat'
[När jag kommer hem] is a subclause, so the subject goes before the verb.

If there is an inte, it also goes before the finite verb – this is called the BIFF rule: bisats inte före finita verbet (a finite verb is a verb that shows time: in Jag tycker om att läsa 'I like to read', tycker om is in the present tense, but att läsa is the infinitive, which does not change depending on time).

Om jag inte kommer hem, ring polisen! 'If I don't come home, call the police!'

3.1 Subordinating vs coordinating conjunctions

Conjunctions can be either coordinating (join two main clauses) or subordinating (join subclause and main clause).

If we compare sentences with subordinating/coordinating conjunctions, they can look like this:
COORDINATING Jag vet och du vet ('I know and you know')
SUBORDINATING Jag vet att du vet ('I know that you know')

Those two look like things are in the same order, so it may seem that it doesn't matter much which one is which. But the difference shows when the BIFF rule comes into play. In a subclause, inte must go before the finite verb. So if we add inte everywhere, we get this:

COORDINATING Jag vet inte och du vet inte heller ('I don't know and you don't know either')
SUBORDINATING Jag vet inte att du inte vet ('I don't know that you don't know')

In the main clauses, we still have V2 and inte gets to go after the verb. In the subclause, inte sneaks in before the finite verb.

3.2 Indirect questions

Indirect questions are more common in the written language than in speech. They often appear after expressions like jag undrar 'I wonder' or jag skulle vilja veta 'I would like to know'.
They are formed like subclauses: the subject goes before the verb (and if there is an inte, it also goes before the verb).
Jag undrar var du är 'I wonder where you are'

Read more about indirect questions here (in Swedish): http://www.student.umu.se/under-studietiden/studieverkstad/skriftliga-uppgifter/skrivrad-och-sprakhjalp/ordfoljd-i-indirekta-fragor/

4. Commands & requests (the imperative)

Much like in English, commands & requests usually have the verb first:
Läs en bok! 'Read a book!'
Unlike e.g. Norwegian, inte always goes after the verb here:
Läs inte den där boken! 'Don't read that book!'

If the person or entity you're addressing is present in the sentence, they can be either first or second, just like in English:
Hjälp mig, mamma! or Mamma, hjälp mig!
'Help me, mom!' or 'Mom, help me!'

5. Exclamations

There is one more type of sentence where the V2 does not apply, exclamations such as Vilken fin bil! 'What a nice car!'

These expressions start with one of the following pronouns or adverbs: så, vad, sådan, vilken. The only thing that can go before them in the sentence are prepositions and indefinite articles.

The shortest version would be like:
Vad hon sover! 'How she sleeps!'

After the pronoun or adverb, there may also follow an adverb that can be compared for degrees:
Vad fort du springer! 'How fast you run!'

There can also be an adjective or participle phrase at the beginning:
Så skickligt hon hanterar problemet! 'How skilfully she handles the problem!'

… or a nominal phrase:
Vilka fina byxor du har! 'Such nice pants you have!'

So, the general structure of this kind of sentence is to put the astonishing thing first, then the subject, then the verb.

(this section is a simplified description based on SAG 4 p. 761-2 'expressive main clauses')