Horns and Hoofs ("Рога и копыта"), an ironical placeholder name for a business engaged in shady or dubious activity. A company with this name was established by Bender to make things look official.
Sitz-Chairman (зиц-председатель, zits-predsedatel), a strawman chairman. The Horns and Hoofs was headed by Sitz-Chairman Funt. The title is a bilingual Russian-Yiddish pun. The Yiddish word "sitzen" means "to sit", which in Russian connotes "doing time". Also "Sitz" has legal meanings similar to the English "seat". The sole function of a Sitz-Chairman was to do prison time when (not "if"!) the time comes. One notable modern usage: "Зицпредседатель" is the Russian title for the film The Hudsucker Proxy.
Children of Lt. Schmidt (Дети лейтенанта Шмидта), a term for small-time con artists. In the book, it was found that there are thirty men and four women impersonating the offspring of Pyotr Schmidt, in order to get money from the government.
Beer is served only to members of the trade union ("Пиво отпускается только членам профсоюза"), an enduring parody of the Soviet system of privileges.
An automobile is not a luxury, but a means of transportation ("Автомобиль — не роскошь, а средство передвижения"). The phrase, reminiscent of Soviet style propaganda, saw some usage within the Eastern Bloc. More recently (circa 1989), Mircea Dinescu opined, A wife is not a luxury, but a means of transportation (referring to people who took Western spouses in order to emigrate).
Keep on sawing, Shura, keep on sawing! ("Пилите, Шура, пилите!"). This ironic phrase refers to an enterprise which is about to fail, especially when continued effort only serves to postpone the inevitable moment of disaster and punishment - a situation known as "death march" in software development. In the novel, two hapless crooks stole kettlebells thinking they had gold cores. The original text in the book omits "Shura", but popular versions usually add the name as above, or alternatively: Keep on sawing, Shura, they are surely golden!
Now I will have to become a building superintendent! ("Придется переквалифицироваться в управдомы" - the last line of the book). Spoken after one's dreams have been crushed and harsh reality is setting in.
No, this is not Rio de Janeiro. ("Нет, это не Рио-де-Жанейро"). Used to describe anything that isn't quite all it's cracked up to be. In the book, it is Ostap Bender's hint to his dream to get rich and move to Rio de Janeiro, to walk in white pants under the bright sun.