Reddit reviews Schaum's Outline of German Grammar, 4ed (Schaum's Outline Series)

There are four grammatical cases in German: Nominative, Accusative, Dative, and Genitive - these are pretty much equated with, respectively, the subject, direct object, indirect object, and possessive in English.

The nominative forms of the definite articles (der, die, das) and the indefinite articles (ein, eine, ein) that indicate gender will change to indicate what the role of each element in the utterance is:

-'Der Apfel ist rot.' - the apple is the subject and, therefore, the nominative 'der' is used.

But:
-"Ich kaufe den Apfel" - the apple becomes the direct object, as it is being acted upon by the subject, 'ich'. the nominative masculine form 'der' has been inflected to the accusative masculine form 'den'.

Then:
-"Ich gebe ihm den Apfel" - now, the apple that is being acted upon, through the act of giving remains in the accusative as the direct object. But we now have an indirect object in 'ihm' the dative masculine form of the nominative 'er'.

Now, this is just a basic gloss of what the accusative and dative cases functions are, but it should answer what you've asked.

Since you didn't ask about genitive I'm not getting into that, as getting the accusative and dative down can be a task in itself.

Here are a couple of links to great resources for grammar:

Schaum's Outline of German Grammar

Also, English Grammar for Students of German

Hi.
If you're looking for some kind of textbook, I've found the Schritte International Glossary XXL German-English books to be quite good when starting out. The first 4 books cover the A1 and A2 levels and are available here. If you'd like, I could send the first volume your way so you could get a feeling for the material.
Otherwise, as an all-in-one solution, I've read good things about Schaum's Outline of German Grammar.
Dictionaries - I don't own a physical one. Online, I use dict.cc and pons.com for word definitions and linguee.de for usage examples.
If my answer is unsatisfactory, also try asking in /r/German which is the dedicated sub-reddit for learning German.

For beginners: http://www.amazon.com/English-Grammar-Students-German-Learning/dp/0934034389/ref=sr_1_2?s=books&ie=UTF8&qid=1413161749&sr=1-2&keywords=german+english+grammar

Beginning to intermediate: http://www.amazon.com/Schaums-Outline-German-Grammar-4ed/dp/0071615679/ref=sr_1_6?s=books&ie=UTF8&qid=1413161727&sr=1-6&keywords=german+grammar

Beginning to intermediate: http://www.amazon.com/Hammers-German-Grammar-Usage-Edition/dp/1444120166

Intermediate to advanced: http://www.amazon.com/Practice-Grammar-German/dp/3190272557/ref=sr_1_1?s=books&ie=UTF8&qid=1413161823&sr=1-1&keywords=hueber+practice+grammar+german

I used Schaums, it's really easy to follow, has loads of examples and you just fill in the blanks over and over until you get it!