In the United States, our smallest circulating coin is the dime (10 cents). Its diameter is 18mm (a little less than 3/4 of an inch), which is only 1mm smaller than the penny (1 cent). The size difference is not easily noticed, which is why the dime is a different color (silver) and has a reeded edge to make the 2 coins more distinguishable.
While the dime seems pretty small, there have been plenty of modern world coins which are even smaller. The main reason to mint a small coin is, of course, to use less metal (which makes the coins less expensive to make). But from a practical sense, a country probably can't make all of its coins small because it would be more difficult to tell them apart. And smaller coins are easier to lose, which is why typically only the lowest denomination (value) coins which are made small.
There is also a practical limit to how small a coin can be. It has to be big enough to be able to read the information on it (year, country, value) and heavy enough to not blow away in a light breeze.
The prize for smallest circulating coin issued since 1900 probably goes to Panama's 2 1/2 centismo, which was issued in 1904 and then again from 1975-1982. This coin was only 10mm wide (almost half the size of a US penny), and due to its tiny size received the nickname "Panama Pill". A 10mm coin seems like it would be difficult to use and easy to lose. In 1907 a new 2 1/2 centismo was minted which measured in at a whopping 21mm (in comparison).
Here is a list of many of the small coins which have been issued around the world since 1900. If you know of anything missing from the list, let us know in a comment.
Algeria 1 centime
Argentina 1 centavo
Armemia 10 luma
Aruba 5 cents
Australia 1 cent, 3 pence
Azerbaijan 1 qepik
Bahamas 1 cent
Bahrain 25 fils
Bangladesh 1 poisha
Belgium 25 centimes
Bolivia 2 centavos
Brazil 5 centavos, 20 reis
Brunei 5 sen
Bulgaria 1 stotinki
Cambodia 50 riels
Costa Rica 5 centimos
Ceylon 1 cent
Chile 5 centavos, 1 peso
China 1 fen
Croatia 1 lipa
Cyprus 1 cent
Czech Republic 10 haleru
Finland 1, 10 penni
Denmark 1, 5 ore
Egypt 1/10 qirsh
Euro 1 cent
Guatemala 1/4 real
Iceland 10 aurar
India 10 paise
Japan 1 sen
Kazakhstan 1 tenge
Mexico 5, 10 centavos
The Netherlands 1/2, 10 cents
Netherlands Antilles 1 cent
Norway 10 ore
Pakistan 1 pie
Panama 2 1/2 centismo
Philippines 1 sentimo
Poland 1 grosz
Romania 5 bani
Russia 1 kopek
Singapore 1 cent
South Africa 1 cent
Spain 1 peseta
Thailand 25 satang
Turkey 1 kurus
United Arab Emirates 1 fil