First let's define some key terms:

Clause: A unit of sentence structure containing a verb, and usually also a subject.

Relative Clause: A clause that particularises or gives information on something or someone. These normally begin with a relative pronoun (who/whom/whose/which/that).

Object: A word or group of words describing what receive or is affected by the action of a verb:

- She ate all the plums.

Plums is the object.

Subject: A word, phrase and so forth that carries out the action of the verb:

- He took the pictures.
- The bridge is very long.

He and bridge are the subjects.

Pronoun: Any word used in place of a noun (I, you, he, she, etc. but also me, who, whom, these etc.)

Relative Pronoun: A pronoun referring to a noun in another (preceding) clause.

Que and Qui in French are relative pronouns used to introduce relative clauses in the same manner that Who/Whom, Which and That do it in English, but their respective grammatical usages follow different regulations.

Like Which and That, Que and Qui have the same meaning, but while Which and That are just about grammatically interchangeable in English (except if the relative clause is independent, in which case Which must be used) Que and Qui are not in French.

Qui

Qui is solely used if it refers back to the direct subject of the sentence:

- La rose qui est dans le jardin.
- The rose that/which is in the garden.

Here Qui refers to rose as the subject of the verb to be.

- La famille qui vous a accueillie.
- The family that/which took you in.

Qui refers to famille, which is the thing doing the action of taking you in.

Que

Que, on the other hand, is used exclusively if it refers back to the object of a sentence:

- Voici le bus que j'attendais.
- Here comes the bus that/which I have been waiting for.

Que refers to bus, which was waited for (and is thus the receipt point of the verb).

- Il veut le verre que tu as nettoy?.
- He wants the glass that you cleaned.

Here Que refers to glass, which is the direct object of to clean.

Elision of 'e' in Que

A important distinguishing characteristic is that with Que, the 'e' is elided (omitted/shortened) and replaced by an apostrophe when the 1st letter of the following word is a vowel. This isn't the same with Qui:

- La bouteille qu'il a ramen? de la cave.
- The bottle that/which he brought back from the cellar.

- La bouteille qui a ?t? ouverte.
- The bottle that/which has been opened.

Qui/Que vs Who/Whom

The other important difference in the use of Que and Qui is that, unlike English, French does not make a difference between living things and inanimated objects (people and things) in the use of relative pronouns.

Whilst Que and Qui can be used to refer to people, in English you can only use Who or Whom for this:

- Le conducteur qui vient de passer
- The driver who just passed by

- Les amis auxquels je pense
- The friends whom I think of

('Auxquels' is a plural form of 'Que' in French. The singular of 'Auxquels' is '? qui' - 'to whom'. Observe the root 'que' in 'auxquels'.)

One cannot state:

- The kid which/that crossed the street
- The minister which/that I helped

Who/Qui versus Whom/Que

Nevertheless, while English does not discern between the subject or object of the sentence when using That and Which, it does when referring to people. Whom, of course, is used when referring to the object, and who is used when referring to the subject of the sentence:

- He is the man who won the first prize.
- Il est l'homme qui a gagn? le grand prix.

Who here refers to man, and man performs the action of winning (and is therefore the subject), whereas:

- This is the person whom I met
- C'est la personne que j'ai rencontr?.

Who here refers to person, and he is the object (receiver) of the verb to meet.

Note: Qui as an interrogative pronoun also means Who:

- C'est qui ? (or, more correctly in French: "Qui est-ce?")
- Who is it?

In conclusion, unlike English where this can now and then be done, the relative pronoun in French can never be taken out:

- C'est la bague je veux.

doesn't make any sense in French (the correct structure is "C'est la bague que je veux"), whereas:

- This is the ring I want.

is quite acceptable in English.

As there are quite some differences, it is always a good idea to have a good grammar and dictionaries available when writing or just translating in a another language. But nothing replaces the help of a native as well. As for businesses, having worked on those lines personally I know of no better value than professional translation services to effectively reach foreign markets.