This is based on many sahih narrations like this in Sahih Muslim:
There are certain ejaculations, the repeaters of which or the performers of which after every prescribed prayer will never be caused disappointment:" Glory be to Allah" thirty-three times." Praise be to Allah" thirty-three times, and" Allah is most Great" thirty-four times.
However the more known practice is saying:
Subhan Allah 33 times.
Alhamdulillah 33 times.
Allahu akbar 33 times.
And completing 100 by saying La ilaha illa Allah (there's no other deity but Allah) as most narrations quote the 99 first words (see for example in Sahih al-Bukhari and al-Muwatta') . And that one linked above saying one should say Allahu akbar 34 times is only a single narration.
All these words are dhikr you praise Allah by subhan Allah, you thank Him by Alhamdulillah and you witness His greatness by Allahu akbar. Note that you may say them as in three sequences of different dhikr (in the order above without mixing praising Allah and thanking him or witnessing His greatness) or as 33 sequences of three different dhikr (subhan Allah, Alhamdulillah, Allah akbar 33 times).