The party or coalition that gets the most votes during the first round also gets the most seats

Any party or coalition has an absolute majority (>50% of seats)

I (Mich) happen to be physically in two different cities for first and second round

Renaissance (Emmanuel Macron's party) gets the highest number of seats but Gabriel Attal doesn't remain Prime Minister

Renaissance (Emmanuel Macron's party) has more seats in the new parliament than in the previous one

The Front Populaire coalition gets a relative majority

Rassemblement National gets a relative majority

The next PM is not from the party / coalition with the most seats 

Jordan Bardella is nominated Prime Minister by Emmanuel Macron but he declines

Emmanuel Macron just announced the dissolution of the Franch parliament. A new parliament is soon to be elected, and as a consequence a new Prime Minister should be designated.

 https://edition.cnn.com/2024/06/09/europe/macron-dissolves-french-parliament-europe-intl/index.html

Anticipated elections are supposed to happen on June 30th and July 7th. The way French Parliament elections work is roughly the following : France is tesselated in a lot of districts, each districts have two-rounds elections where the winner becomes a member of parliament. Normally, the Prime Minister is from the party that won the most seats.