The genitive case has four main uses:
Counting - When counting objects ("I have six sheep"), the thing being enumerated is usually put in the genitive case.
Possession - If something is owned by something else, the owner is in the genitive case. This is similar to the 's' in English: "my brother's car". "Brother's", in Russian, would be the singular word "brother", брат, in its genitive form, брата. This also corresponds with the English 'of', as in 'car of my brother'.
Negation - To say something is there, we generally use the nominative case. If it isn't there, we use нет followed by the genitive.
Prepositions - A number of common prepositions call the genitive case. The most common is у, which means 'by' or 'near'. It's also used to say "I have...": У + gen есть + nom.
The accusative case is used for the direct object of a verb. In "Bob eats lunch," "lunch" is the direct object. In English, the object of a sentence is indicated by context and word order, with a strict subject-verb-object (SVO) system. In English, "Bob eats lunch" and "Lunch eats Bob" have different meanings. But in Russian, since the two cases are distinguished by suffixes, we can change the order of words and still know who's the object and who's the subject. If English indicated the direct object by adding "oo", we could say "Bob eats lunchoo" or "Lunchoo eats Bob", and either way it would clear that Bob was doing the eating.
One nuance of the accusative case is that it makes a distinction of animacy. That is, masculine nouns denoting people or animals (i.e., animate nouns) take their genitive form, while neuter nouns and inanimate masculine nouns take their nominative form. Plural nouns of any gender take their nominative plural form if inanimate, or their genitive plural form if animate.
The instrumental case is used to denote the object by which something is done. In the sentence, "I am writing with a pencil", the phrase 'with a pencil' in Russian is simply the word 'pencil' (карандаш) put into its instrumental form (карандашом). In this way, it corresponds to one of the two meanings of the word 'with' in English: 'with', as in 'by means of'. The other meaning, 'in the company of', such as "I rode with Jane", is also commanded by the instrumental case, though this requires a preposition: с + inst.
There are five other common prepositions that command the instrumental: под (pod, 'under') над (nad, 'above'/'on top of') за (za, 'behind') между (mezh-doo, 'between'), and перед (pye-red, 'before'/'in front of').
The prepositional case is also called the locative case, as it was commonly used to denote a sentence's object's location or an activity. Nowadays, however, it's only ever called after a few prepositions, hence its name. It's not rare, though, and the prepositions that call it are the most common of all Russian prepositions. They are в (in), на (on), and о/обо (about; it's обо in the phrase "about me" or "about my"). However, these prepositions can also call other cases, in which case their meaning changes. For instance, в + prep means 'in', as in, "I live in England". в + acc, however, means 'into' or 'to', as in, "I'm flying to New York".