Let's start with the easy one: analysis. This is the method English uses for its core roles: subject, direct object, and sometimes the indirect object. As I pointed out in The Decline of the English Language, Modern English has a fairly rigid word order for its core roles: Subject Verb [IndirectObject] [DirectObject], as in "The boy gave the dog a bone"; some other word orders are used by native speakers, but they're uncommon, and generally only used in certain specific contexts (e.g. the Verb Subject Complement order in "Are you an idiot?"). Thus analysis refers to the use of strict word ordering to determine what role each noun has.
As I mentioned in the same paper, English wasn't always this way: it belongs to the same language family as Latin, all traditionally using dependent-marking of case. Dependent marking refers to the fact that each word is marked to indicate its role. In the same sentence "Puer [boy] cani [dog] os [bone] dabat [gave]", the four words may be placed in any order, and the meaning will still be clear, because the nouns carry the nominative, dative, and accusative cases, respectively (actually, that isn't 100% true; because some cases decline the same way, there can be some ambiguity here).
You might notice that English also does this for non-core roles, which corresponds to greater freedom as to word order. As dependent-marking does not require that the mark actually be attached to the word, English uses prepositions to mark non-core roles, rather than the traditional suffixes of Indo-European languages. This system is used for such roles as instrument in "The boy poked the dog with a bone" (the Latin version, "Puer canes osse pungebat", uses the ablative case, and the accusative case for the dog), the benefactor in "The boy bought a dog for her" (in the Latin version "Puer canes per ea emebat", a preposition is used with the ablative in this case), etc. The last example also illustrates that Latin uses prepositions as well, to mark roles outside the 6 core cases.
Both of those have been something that isn't entirely unfamiliar to English speakers. Even case still (barely) exists in the pronouns and nouns of English (having three and two cases, respectively); the third method, head-marking or agreement, is also not absolutely foreign, though it is uncommon in modern English. Verbs in Indo-European languages traditionally agree with the subject of the sentence - the verbs themselves indicate the grammatical person and number of the subject. While English has all but lost this form of agreement, you can still see vestiges of it. The verb 'am' uniquely identifies the subject as first person singular, while 'is' identifies the subject as third person singular ('are' is ambiguous, because it could refer either to second person singular or any person plural); similarly, the -s form of all other verbs (e.g. 'gives') identify the subject as third person singular. Romance languages like Spanish still contain robust subject-verb agreement, such that it is possible to uniquely identify the subject as first, second, or third person (never mind the bad terminology for now) and singular or plural.
However, you might have noticed something: in languages like Latin that have subject agreement, marking nouns with the nominative case (used for the subject) can be redundant. Head-marking, or polysynthetic, languages do away with this use of case, and purely rely on verb agreement to indicate which nouns have each role. I can't find a good example of a sentence that would indicate how this would work without introducing other things I don't want to get into, so I'm gonna make one up:
In this example, theyare attachedit pronouns representingthem the subject, direct object, and indirect object the verb of each clause. As with English pronouns in general, theyagree the attached pronouns with number and gender of the nouns. For the verbs, iusedthem the subject-verb-object order and pronoun cases, to makethem the verbs easier to read for English speakers. However, iusedthem varying word orders for nouns in the clauses to illustrateit how itcan be used head-marking with different word orders. Typically theywould useit head-marking head-marking languages with other modifiers like possessives, as well.Finally, the Totonac language takes polysynthesis to a ridiculous extreme. According to the examples in The Grammar of Discourse, Totonac merely lists all roles in the sentence, without using agreement to indicate which nouns have which roles. One example given (I'm kind of making up my own orthography, here) is "liiteemaktamaahua [literally 'with-passing by-from-buy'] tumin [money]", which means "As [he] passes by, [he] buys [it] from [him] with money". Amazingly (and completely against expectations), native speakers of Totonac can actually understand each other.