As the self-accepted answer is outdated and there’s a follow-up question, I’m posting a new answer.

You can find all the visa information by visiting the website of the Ministry of Foreign Affairs of Azerbaijan and going through the “CONSULAR SERVICE AND VISA” – “Visa” tabs.

Here you will find the link to List of states whose citizens can be issued visas at international airports of the Republic of Azerbaijan. Here you will see that China is among the countries whose citizens can be issued visas upon arrival:

The list from that link:

    Name of the country         Validity period of issued visa

  1 Qatar                       30 days
  2 Oman                        30 days
  3 Saudi Arabia                30 days
  4 Bahrain                     30 days
  5 Kuwait                      30 days
  6 Japan                       30 days
  7 Indonesia                   30 days
  8 People`s Republic of China  30 days
  9 South Korea                 30 days
 10 Malaysia                    30 days
 11 Singapore                   30 days
 12 Republic of Turkey          60 days
 13 Israel                      30 days
 14 United Arab Emirates        30 days

Note: Only citizens of USA traveling with New York – Baku flight can
be issued 30 days visa at Heydar Aliyev International
Airport.

Regarding your follow-up question:

My friend who was on the Tbilisi-Baku train was not permitted by the Georgian customs and immigration people to leave at this border
crossing. Why?

Again, in the link provided above, there’s another link to Documents required for visa issuance. In the bottom of the page there’s the following section:

Following documents shall be submitted for visa issuance at a visa
section of the Consular Department of the Ministry of Foreign Affairs
of the Republic of Azerbaijan at the International Airport named after
Haydar Aliyev; visa sections of the Ministry of Foreign Affairs of
the Republic of Azerbaijan at the International Airport in Nakhchivan
Autonomous Republic, at Ganja International Airport; at Lankaran
International Airport; at Gabala International Airport; at Zagatala
International Airport of the Republic of Azerbaijan:

• 1 application form (sample is attached – download link);
• A passport of a foreigner and an identity card of a stateless person;
• A receipt about payment of state duty for visa issuance.

I can’t tell exactly why Georgian customs didn’t permit your friend to enter Azerbaijan, but if your friend didn’t have any legal problems with leaving Georgia and the immigration people stated explicitly, that your friend needs a visa to Azerbaijan, then from the link I’ve referenced above, it might be that your friend needed the aforementioned application form (Application form to obtain a visa in the territory of the Republic of Azerbaijan) or this application form is issued and accepted only at the international airports of Azerbaijan.

Long story short, I can only assure you, that 2 of my Chinese friends visited Baku and both got the visa upon arrival. The catch is that they came directly to Baku by plane, as expected from most of the Chinese tourists visiting Azerbaijan.

Officially it is “visa free” according to WIKI Travel page.
I’m not sure whether it is a mistake made by them or not but here http://bbs.qyer.com/thread-1044622-1.html
seems in practical Chinese nationals still need a visa for Azerbaijian.

It turns out the information is indeed on the Azerbaijan Ministry of Foreign Affairs website, just the link does not stand out well and the page contains at least one other broken link.

The countries applying the visa-free regime page has a section

II group.
For the holders of any kind of passports

(Diplomatic, service, (official/special) and ordinary)

Mutual visa-free regime is applied between the Republic of Azerbaijan and countries listed below for
1. National of both countries holding any kind of valid passport –including diplomatic, service (official/service) or ordinary passports providing the right of international cross-border;
2. Family members (supposes and children) of above mentioned persons:
– traveling with them;
– having no personal passports;
– photos, given name, family name and patronymic officially registered
in relevant type of passports
3. stateless persons holding relevant identity document providing the right of international cross-border and granted especially to the stateless persons by the relevant state bodies of the Azerbaijan Republic or appropriate foreign country;
4. Nationals or stateless persons holding Reentry Certificate – temporary document for international cross border, granted only for returning to the citizenship country or permanent residence country in the cases of lost/theft/became invalid/expiry of passport.

The heading of this section is in fact a link, which I missed. It links to a PDF which was Updated: 21.10.2010 and contains:

No Name of the country The allowed duration for staying in
                       the territory of both countries
                       without visa

1  Belarus             90 days
2  China               30 days
3  Georgia             90 days
4  Kazakhstan          90 days
5  Kyrgyzstan          90 days
6  Moldova             90 days
7  Mongolia            90 days
8  Russia              90 days
9  Tajikistan          90 days
10 Ukraine             90 day
11 Uzbekistan          90 days

UPDATE

Despite all this evidence, my friend who was on the Tbilisi-Baku train was not permitted by the Georgian customs and immigration people to leave at this border crossing, despite him having printouts of the information I listed above!

If anybody can suggest why this occurred it might well make a better answer! \-: