königlich
royal
Relating to or characteristic of a king, queen, or monarchy.
Die königliche Familie besuchte die Eröffnungszeremonie.
The royal family attended the opening ceremony.

royal
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royal
Relating to or characteristic of a king, queen, or monarchy.
Die königliche Familie besuchte die Eröffnungszeremonie.
The royal family attended the opening ceremony.

The adjective 'königlich' (royal) is derived directly from the noun 'der König' (the king). The ending '-lich' is often used to form an adjective from a noun.
Das königliche Wappen ziert den Eingang des Palastes.
The royal coat of arms adorns the entrance to the palace.
In its literal meaning, 'königlich' (royal) describes nouns that are directly connected with a monarchy. Common combinations are 'die königliche Familie' (the royal family), 'das königliche Schloss' (the royal palace/castle), or 'königliche Hoheit' (royal highness).
Die Kronjuwelen werden im königlichen Palast aufbewahrt.
The crown jewels are kept in the royal palace.
In the figurative meaning, the phrase 'sich königlich amüsieren' is very common and means that you are having a lot of fun. The word can also intensify other verbs of enjoyment, such as 'königlich speisen' (to dine like royalty).
Die Kinder haben sich auf dem Geburtstagfest königlich amüsiert.
The children had a wonderful time at the birthday party.
regal
'königlich' and 'regal' can both express that something is fit for a king or queen, or that it seems very majestic. 'königlich' is the normal word in German and fits in many situations, for example with palaces, families, or ceremonies. 'regal' is rare and sounds elevated or literary. It is used more when you want to describe a particularly dignified or ceremonial impression. In everyday language, 'königlich' is almost always more natural.

Im Museum sieht die Klasse eine königliche Krone aus Gold.
At the museum, the class sees a royal crown made of gold.
Die Schauspielerin stand in einem langen Mantel vor dem Publikum und wirkte sehr regal.
The actress stood before the audience in a long coat and looked very regal.