rufen
to shout
To say or shout something in a loud voice in order to be heard.
Sie musste laut „Hilfe“ rufen, um auf sich aufmerksam zu machen.
She had to shout 'Help!' loudly to attract attention.

to call
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to shout
To say or shout something in a loud voice in order to be heard.
Sie musste laut „Hilfe“ rufen, um auf sich aufmerksam zu machen.
She had to shout 'Help!' loudly to attract attention.

The preposition 'nach' + dative is used to express that one wishes for or needs someone or something. 'Nach jemandem rufen' often means that you miss the person or need their help.
Das kleine Kind ruft nach seiner Mutter.
The small child is calling for his mother.
When you want to reproduce the exact words someone shouts, you often use a direct quote in the sentence. A very common fixed phrase is 'um Hilfe rufen' (to call for help).
„Vorsicht!“, rief er mir noch zu.
'Careful!' he still shouted to me.
When you ask a person to come by saying their name, that person is in the accusative. This is the most common construction in everyday life.
Ich rufe die Kinder, das Essen ist fertig.
I'm calling the children; the food is ready.
In the meaning 'bestellen' (to call/order/summon), 'rufen' is almost always used for emergency services or means of transport. Typical combinations are 'die Polizei rufen' (to call the police), 'die Feuerwehr rufen' (to call the fire department), 'einen Arzt rufen' (to call a doctor), or 'ein Taxi rufen' (to call a taxi).
Wegen des Rohrbruchs mussten wir einen Handwerker rufen.
Because of the burst pipe, we had to call a tradesperson.
The phrase 'ins Leben rufen' is a fixed expression. It is used for founding something new and positive, such as a project, an organization, or an event.
Die Bürger haben eine neue Initiative ins Leben gerufen.
The citizens launched a new initiative.
to say
'sagen' and 'rufen' both have to do with speaking. 'sagen' is general: you express words, information, or an opinion. The volume is not important. 'rufen' means that you say something loudly so that others can hear it, or that you want to call someone over to you. You can also 'einen Arzt rufen' (call a doctor), 'die Polizei rufen' (call the police), or 'ein Taxi rufen' (call a taxi). Learners may confuse the words because in both cases someone is speaking. The most important difference is the situation: 'sagen' fits normal messages or statements, while 'rufen' fits a loud voice, distance, or calling someone or something to come.

Im Park ruft die Mutter ihr Kind, weil es zu weit weg läuft.
In the park, the mother calls her child because it is running too far away.
Nach dem Unterricht sagt der Lehrer, wann die nächste Prüfung ist.
After class, the teacher says when the next exam is.