In a way, marionettes seem like the most logical of puppets: basically jointed dolls with strings attached to their limbs. There was a renaissance of marionettes in Britain in the first half of the last century; one of George Bernard Shaw's final plays was written for the Malvern Festival of puppets. Marionettes are an important part of the revival of Czech culture under the Austro-Hungarian Empire in the 19th century. They continue to be important in the Czech performing arts, and Czech marionette troupes are found not only in Prague, but also in New York City.
In Western culture, the most single most famous puppet character is Mr. Punch, of Punch and Judy. He is in many countries: in Russia, he's Petrushka; in the Netherlands, Jan Klaassen; in Romania, Vasilache. In the middle of the last century, certain puppets became famous on television: Miss Piggy, Howdy Doody, Madame (of Wayland Flowers and), Lampchop, Ollie the Dragon, etc. One internationally famous puppet is Topo Gigio; originally on Italian television, and appeared frequently on the Ed Sullivan Show.
There are many puppeteers who left their mark on the world of puppets, and whose characters are embedded in our minds: Jim Henson, Bil Baird, Shari Lewis, etc. Burr Tilstrom created Kukla, Ollie and the rest of the Kuklapolitan Players (their Christmas card is above). Puppeteers in Paris risked imprisonment during the Revolution if they performed. Bil Baird and George Latshaw were puppets who performed live and on television all over the United States during the second half of the last century. Although I once hear Austin Pendleton describe working on a Muppet movie as "spending all day shooting so that Kermit's string doesn't show." I’ve also read Edward Albee’s description of going over to Burr Tillstrom’s place and he’d only address Albee in character as Beulah Witch (which becomes weirder the more you think about it). The link above is to a Tillstrom site which itself has some wonderful links, as well as the author’s first hand impression of him.