cigarette
The German word for cigarette is 'Zigarette'. It is used in the same way as the English word, to refer to a thin cylinder of finely cut tobacco rolled in paper for smoking. In German, it is a feminine noun and is used in the same contexts as in English.
I smoke a cigarette after lunch
The speaker is saying that they have a routine of smoking a cigarette after they have their midday meal.
Can I please have a cigarette?
This is a polite way to formally request a cigarette from someone. The speaker in this example could be at a social event and asking a friend or stranger for a cigarette.
Cigarette, ashtray and matches are all on the table
This phrase describes a scene where both a cigarette, a receptacle for ashes (ashtray), and a box of matches (which can be used to light the cigarette) are located on top of a table.