das Handy
mobile phone / cell phone
A portable, cordless mobile telephone used for mobile communication.
Mein Akku ist leer, ich muss mein Handy aufladen.
My battery is dead; I need to charge my cell phone.

mobile phone / cell phone
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mobile phone / cell phone
A portable, cordless mobile telephone used for mobile communication.
Mein Akku ist leer, ich muss mein Handy aufladen.
My battery is dead; I need to charge my cell phone.

'Das Handy' (mobile phone/cell phone) is a typical example of 'Denglisch'. Although it sounds English, it is not the English word for a mobile phone. In English, you say 'mobile phone' or 'cell phone'. The English adjective 'handy' means 'practical' or 'useful'.
Darf ich mal dein Handy benutzen? Mein Akku ist leer.
Can I use your cell phone for a second? My battery is dead.
In everyday life, 'Handy' (mobile phone/cell phone) is combined with many typical verbs. These include: 'das Handy laden/aufladen' (to charge the phone), 'am Handy sein' (to be on the phone) or colloquially 'am Handy hängen' (to be glued to one's phone), 'aufs Handy schauen' (to look at one's phone), or 'eine Nummer im Handy speichern' (to save a number in one's phone).
Er ist ständig am Handy und hört gar nicht zu.
He's constantly on his phone and isn't listening at all.
'Handy' (mobile phone/cell phone) is a very productive word and forms the basis for many useful compounds. Examples are 'Handynummer' (cell phone number), 'Handyhülle' (phone case), 'Handyvertrag' (cell phone contract), or 'Handy-Akku' (phone battery).
Ich brauche eine neue Handyhülle, meine alte ist kaputt.
I need a new phone case; my old one is broken.
telephone
'das Handy' and 'das Telefon' are both devices you can use to speak with other people. 'Handy' means a mobile phone that you carry with you. 'Telefon' is more general and can mean a cell phone, but also a landline phone in an office or at home. Learners may confuse the words because you make phone calls with both. If you mean the device in your pocket, 'Handy' is very natural. If the type of device is not important, or if it is a device on a desk, 'Telefon' fits better.

Im Zug sucht Lisa ihr Handy, weil sie eine Nachricht schreiben will.
On the train, Lisa looks for her cell phone because she wants to write a message.
Auf dem Schreibtisch klingelt das Telefon seit einer Minute.
The telephone on the desk has been ringing for a minute.