'Agreement' refers to elements in a sentence having the same number, gender, case or person. In English, it's probably an issue only for number (that is, singular vs plural) and case (that is, 'I' vs 'me', 'he' vs 'him' and so on).

Agreement in number

Probably the commonest cause of disagreement in number is using 'they' and 'their' for a gender-neutral singular, as in: Each student supplies their own books. This is increasingly being accepted, but it's rarely necessary. Making the noun plural - the students supply their own books - usually works. If it doesn't, the occasional 'his or her' does no harm.

Sometimes, you can avoid the problem by rewriting. If the employee is unhappy with the judgment, they can appeal to the Human Rights Commission can easily become: An employee who is unhappy with the judgment can appeal to the Human Rights Commission. (All the same, wouldn't a gender-neutral pronoun be great? Maori have the right idea, with the word 'ia' - meaning 'he, she or it'.)

Mixed number is also common with collective nouns like 'group', 'team' or 'couple'. Because these things involve more than one individual, we tend to use plural verbs with them - but really, they are entities in themselves and are therefore singular. So ... 'the group', 'the team' or 'the couple' 'is', not 'are'; 'has', not 'have'.

Sometimes the reason for the disagreement is more subtle: Neither of my children take after me. If there's a plural noun immediately before the verb, the verb tends to be incorrectly made plural as well, even though that noun isn't its subject. In this example, the subject is "neither", which is singular: Neither of my children takes after me.

The same sort of thing happens when there is more than one noun: Neither my son nor my daughter take after me. Although there are two nouns, they are alternatives and are therefore, in effect, singular: Neither my son nor my daughter takes after me. Watch out for 'neither', 'none', 'nor' and 'or'.

Agreement in case

Disagreement in case - in English, anyway - is a by-product of a tendency to use the wrong case in certain circumstances.

'Case' refers to the different forms that nouns, adjectives and pronouns can take to show their relationship to other words in the sentence. If that sounds daunting, it helps to know that in English, only two case forms have survived, and only in the personal pronouns. They are the subjective form: I, he, she, we and they; and the objective: me, him, her, us and them. (The other pronouns, 'you' and 'it', have the same form for both cases.) The subjective case is used for the subject of the sentence; the objective case is used after prepositions and for the object.

Errors involving case usually stem from the tendency of the objective case to creep in where it shouldn't, such as in It's me (a use that is now acceptable) for It is I, and You're a better player than him for You're a better player than he (is). The reason disagreement comes into it is that, for some reason, people are more likely to get their cases wrong when they put two pronouns together ... so you get sentences like Them and I did it, and Her and I are friends. (Of course, both cases are wrong sometimes too, as in Me and him are to blame.)

Less often, the subjective form appears where it shouldn't: It's up to you and I, rather than It's up to you and me. This is often due to a phenomenon that linguists call 'hypercorrection'; people have a vague understanding that they sometimes use the objective form when they shouldn't, so they counteract this by not using it when they should.

People rarely get their cases wrong when there's only one pronoun (few would say Him is to blame or It's up to I), so if you have two together and you're not sure if they're right, reading the phrase twice - once for each pronoun - should point up any mistake.