In order to join a public party, you need to deposit a satisfactory amount of the denominated asset (usually USDC). You may not be able to join if the party is full. When you join a party, you will receive a liquidity token, which is your proof of ownership of your stake in the party. Even if more people join before or after you, your share remains the same and is only affected by the price increase or decrease of the assets the party purchases. You can learn more about the technical aspect of how this is acheived here.