American drugstores do provide a pharmacy, but most also sell books,
magazines, small electronics, food, personal care products, toys... In
other words, they are general stores that also fill prescriptions.
What is such a store called in ENglish by Europeans?

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Request for Question Clarification by scriptor-ga on 25 Apr 2006 07:51 PDT

By what Europeans exactly? There are Germans, French, Spanish,
Italians, to name only some ... it's a continent with dozens of
languages.

Scriptor

<We call them chemist shops or chemists. 

Boots is the largest chain that sells a whole variety of goods. 

Although pharmacy means a place where medicines are dispensed,
pharmacy is rarely used in colloquial English.

WWlib gives a list of American and English language uses/spellings.
http://www.scit.wlv.ac.uk/~jphb/american.html

The chemist is where you'd go to buy pharmaceutical drugs. Americans
call it a straight drugstore, which implies to Brits that you could
just buy Class A narcotics over the counter. These days it's
acceptable in both Britain and America to call the place a pharmacy.
We do still call a person who works with chemicals a chemist, as
Americans do. Source: The English to American Dictionary.
http://english2american.com/dictionary/c.html

You can see the use of chemists and chemist shops in this article. OFT
ruling takes fight to chemists.
http://business.scotsman.com/retail.cfm?id=64452003>

<Hope this helps.>

I think they call them Chemists.
"I went down to the Chemists to get some Ex-lax"
Not sure if this is a slang use of the term but just what i've heard.

Maybe in England drugstores offer the broader range of products that
one is accustomed to find in the States behind a Longs or Rexall sign.
 Boots, once "Boots the Chemist", now the Boots Group, is a nationwide
chain and is also vertically structured, not just selling, but also
producing.
http://en.wikipedia.org/wiki/Boots_Group

In Germany  - and perhaps other European countries -  the laws
controlling pharmacists ("Apotheker") not only require that a trained
and licensed pharmacist owns and manages a pharmacy, until very
recently, he could only control a single store.  Now he or she may
have an additional 3 branch stores within the nearer area.  The law
considers a pharmacist to be self-employed porfessional and not only
licenses him for his professional qualifications but also licenses his
operation of a pharmacy.  Thus the US type of drugstore chains could
not develop in Germany.
I didn't find a restriction on the type of goods a German pharmacist
may sell, though there may be one.  It seems that they only sell
prescription and non-prescription items and other things relating to
health care, many of which by law can only be sold in a pharmacy.
My suspicion is that the profit margin on these products is high
enough to make it uninteresting to attempt to sell other goods. 
Besides, by local tradition, people don't expect to find such there.
The German "Drogerie" (literally "druggist", but not a pharmacy) also
sells healthcare items that are not limited to sale by a pharmacist,
i.e., a very broad selection of good for personal care:  make-up, sun
cream, deodorants, etc., etc., and often handle photo development. 
These are self-service and look more like the equivalent section of a
US drugstore.  And they are often chain stores, and a couple of them,
Müller Drogerie and Schlecker, have a much broader range of goods,
from racks of CDs to wooden toys, as this site mentions, speaking of
130,000 products and stores with as much as 30,000 sq.ft.:
http://www.manager-magazin.de/koepfe/unternehmerarchiv/0,2828,306698-3,00.html

These were be the equivalent to a US drugstore, except that they have no pharmacy.

I'm in Australia and we use the term 'chemist' for such a store.  I
imagine this would be the same in Europe