laut
loud
With a high volume; clearly audible and not quiet.
Die Musik auf der Party war so laut, dass wir uns kaum unterhalten konnten.
The music at the party was so loud that we could hardly talk to each other.

loud / according to
Loading your saved state.
loud
With a high volume; clearly audible and not quiet.
Die Musik auf der Party war so laut, dass wir uns kaum unterhalten konnten.
The music at the party was so loud that we could hardly talk to each other.

The word 'laut' has two completely different functions. As an adjective, it describes a high volume. As a preposition with the meaning 'gemäß' (according to), it introduces a source of information and comes before a noun.
Die Musik ist mir zu laut.
The music is too loud for me.
When 'laut' is used as a preposition, the following noun can be in the genitive or dative. In modern language, especially in spoken German, the dative is preferred.
Laut dem neuen Gesetz ist das Rauchen hier verboten.
According to the new law, smoking is prohibited here.
Like many adjectives, 'laut' can also be used as an adverb to describe how an action is performed. In this case, it is not declined and usually comes after the verb.
Er lachte laut über den Witz.
He laughed loudly at the joke.
to / after / according to
'nach' and 'laut' can both express that you are referring to a source, rule, or piece of information. 'nach' is very general and works with opinions, plans, rules, or information. 'laut' is used especially when you name a specific source, for example a report, a news item, or a statement. There is a risk of confusion because both are often explained with 'gemäß' (according to). Remember: 'laut' sounds more strongly like an information source; 'nach' is broader and is also very natural with opinions.

Laut Wetterbericht regnet es morgen den ganzen Tag.
According to the weather report, it will rain all day tomorrow.
Nach Meinung der Ärztin sollte Ali heute zu Hause bleiben.
In the doctor's opinion, Ali should stay home today.