Due to the increase in mobile penetration, public phones are less common that they used to be. They can be found in post offices, railway stations, airport stations and urban pedestrian areas.
In the "good old days", you could easily spot their bright yellow color, but today most Telekom booths have been changed to a "stylish" pink-lavender which is harder to spot.
To call you will probably need a telephone card. Some phone booths still accept coins, but these are very rare, so don't depend on it.
Telekom cards can be bought at the T-Punkte of Deutsche Telekom, post offices, and many stationery shops in denominations of €5, 10, and 15. You can also buy prepaid cards from other providers which can save money and are used by dialing a PIN. Credit cards can be used from a small number of public phones.
In most public telephone boxes it is also possible to be called back, if so the number will be displayed.