It has to do with dependent and independent clauses, and actually I had a grammar class earlier this year about that, but now, as usual, I don't remember anything :(

These sites might be helpful: owl.english.purdue.edu/handouts/grammar/g_clause.html & http://webster.commnet.edu/grammar/commas.htm

According to Strunk and White in The Elements of Style:

"That is the defining, or restrictive pronoun, which the nondefining, or nonrestrictive ...
The lawn mower that is broken is in the garage. (Tells which one)
The lawn mower, which is broken, is in the garage. (Adds a fact about the only mower in question.)

The use of which for that is common in written and spoken language (...) Occasionally which seems preferable to that, as in the sentence from the Bible. But it would be a convenience to all if these two pronouns were used with precision. The careful writer, watchful for small conveniences, goes which-hunting, removes the defining whiches, and by doing so improves his work."

So, if you can leave the clause out without changing the meaning of the sentence, use which preceded by a comma. If you cannot leave it out, use that without a comma.

I had exactly the same experience and looked it up in my English grammar book. There is real choice - something that computers can't always handle. The important thing is obviously to use commas in relative clauses. Which or that may be used when talking about things (as opposed to people)in relative clauses e.g. Where is the cheese that/which was in the fridge?
If anything that is more commonly used in sentences like, "The computer that broke down has now been repaired", but "which broke down" would also be correct.
Also one can say, "Do you know anyone who/that speaks French and Italian?".
You have to use which for things like "Colin told me about his new job, which he is enjoying very much".

So you can sometimes ignore the Spellcheck. Source:English Grammar in Use, Raymond Murphy

I feel I can add little to the excellent answers already given, except this quote from Oxford 'Usage':

Relative clauses can be either restrictive or non-restrictive. A restrictive relative clause serves to restrict the reference of the antecedent, e.g. A suitcase which has lost its handle is useless. Here the antecedent suitcase is defined or restricted by the clause.



A non-restrictive relative clause is used not to narrow the reference of the antecedent, but to add further information, e.g. He carried the suitcase, which had lost its handle, on one shoulder. Here the suitcase is already identified, and the relative clause adds explanatory information.



Notice that no commas are used to mark off a restrictive relative clause from the rest of the sentence, but when, as above, a non-restrictive relative clause comes in the middle of the sentence, it is marked off by a comma at each end. There are two kinds of relative pronouns:



(i) The wh-type: who, whom, whose, which, and, in nominal relative clauses only, what.

(ii) The pronoun that (which can be omitted in some circumstances: see *that (relative pronoun), omission of).


It's interesting that they still suggest the use of 'which' where more modern practice (incl. Word!) prefers that.

If you're writing American English, it's a good bet to rely on US style manuals for correct usage.
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From the New York Times Manual of Style and Usage, 1999:

that, which - Use 'that', not 'which', in a restrictive clause - a clause necessary to the reader's understanding of the sentence: 'The town that the pitcher calls home is tiny Hawley, Pa. (The sentence serves no purpose without 'that the pitcher calls home.')
Note that there are no commas around the clause.

In a nonrestrictive clause - one providing added information, not essential to understand the sentence - use 'which', preceded by a comma: Hawley, Pa., which the pitcher calls home, is tiny.
(The sentence is understandable without 'which the pitcher calls home.'

I have noticed the same phenomenon with the syntax checker on Word.

I think it is wrong.

There are instances where "which" sounds right, and there are instances where "that" sounds right. Sometimes, either is OK, and sometimes one sounds distinctly odd to the native speaker's ear.

My advice to you, if you're a near-native speaker, is to use whichever seems more natural. As long as you observe the distinction mentioned in Kim's excellent answer, you will not be misunderstood or commit a serious solecism.

I would also point out that it is a very common error to put a comma before or after a which-clause, but not both, which is almost always wrong. It is OK to have only one comma if the clause comes at the end of a sentence, which is obvious.

Exactly the same considerations apply to "who" and "that" clauses, of course.

"My sister, who lives in Norway, recently got married."

This suggests that either I only have one sister or I don't feel like mentioning the other(s) and that I am throwing in a little bit of additional but non-essential information about her.

"My sister who lives in Norway recently got married."

If I have just mentioned that I have two sisters, one living Norway and one in Afghanistan, using this construction would serve to identify which sister I am talking about. The "who lives in Norway" is not additional but non-essential information; it is part of the definition of the subject.

I think it is especially important for native speakers of German to observe this distinction, as subordinate clauses are always required to be set off with commas. (This consideration may also be useful interpreting the writings of German speakers who are not expert at English.)

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Note added at 8 hrs 22 mins (2004-09-14 04:21:40 GMT)
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An odd story. My grandmother, who was born in the 1880s, was an English teacher. As I child, I read in one of her even then very old British textbooks that \"which\" was, strictly speaking, the *only* grammatically correct relative pronoun, and that \"who\" and \"that\" were therefore ungrammatical used as relatives. It went on to say, however, that \"the best writers of all generations\" had ignored this rule.

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Note added at 8 hrs 22 mins (2004-09-14 04:21:43 GMT)
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An odd story. My grandmother, who was born in the 1880s, was an English teacher. As I child, I read in one of her even then very old British textbooks that \"which\" was, strictly speaking, the *only* grammatically correct relative pronoun, and that \"who\" and \"that\" were therefore ungrammatical used as relatives. It went on to say, however, that \"the best writers of all generations\" had ignored this rule.

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Note added at 8 hrs 22 mins (2004-09-14 04:21:44 GMT)
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An odd story. My grandmother, who was born in the 1880s, was an English teacher. As I child, I read in one of her even then very old British textbooks that \"which\" was, strictly speaking, the *only* grammatically correct relative pronoun, and that \"who\" and \"that\" were therefore ungrammatical used as relatives. It went on to say, however, that \"the best writers of all generations\" had ignored this rule.

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Note added at 8 hrs 56 mins (2004-09-14 04:55:46 GMT)
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Sorry about the note being repeated above. I must have inadvertently clicked twice.

US vs UK: Kim has expressed the view that there is a US vs UK divide in the use of \"which\" vs the use of \"that\". I think this is only partially true.

While US teachers and syle-book authors may preach the use of \"that\" for restrictive clauses and \"which\" for non-restricitve clauses, many educated Americans, including some noted authors, seem happy to ignore this distinction. When you couple this with the observation that UK style books, such as the one quoted by Dusty, use as examples of correct usage sentences with restrictive which-clauses, it seems hard to justify this so-called rule on either side of the Atlantic.

I note that my installation of Word objects to restrictive which-clauses when set to British or Australian English, as well as when set to US English. I don\'t suppose we can do much about economic imperialism, but I draw the line at this syntactic imperialism.