Here is my take on the matter.
App client has several Auth Flow Configurations.
1. Enable username password auth for admin APIs for authentication (ALLOW_ADMIN_USER_PASSWORD_AUTH)
This enables Server-Side Authentication Flow. If you don't have an end-user app, but instead you're using a secure back end or server-side app.

2. Enable lambda trigger-based custom authentication (ALLOW_CUSTOM_AUTH)
This enables the Custom Authentication Flow. This can help you create a challenge/response-based authentication model using AWS Lambda triggers.
https://docs.aws.amazon.com/cognito/latest/developerguide/user-pool-lambda-challenge.html
Under User Pools -> Triggers you can see many lambda functions. You can use Create Auth Challenge, Define Auth Challenge and Verify Auth Challenge Response functions to create a custom authentication flow.
3. Enable username password based authentication (ALLOW_USER_PASSWORD_AUTH)
This enables Client Side Authentication Flow that uses user password-based authentication. In this flow, Cognito receives the password in the request.
You can use AWS Mobile SDK for Android, AWS Mobile SDK for iOS, or AWS SDK for JavaScript to implement this.
4. Enable SRP (secure remote password) protocol based authentication (ALLOW_USER_SRP_AUTH)
This is similar to the above flow in section 3. except for the password verification. This flow uses the SRP protocol to verify passwords.
http://srp.stanford.edu/whatisit.html
https://docs.aws.amazon.com/cognito-user-identity-pools/latest/APIReference/API_UserPoolClientType.html

5. Enable refresh token based authentication (ALLOW_REFRESH_TOKEN_AUTH)
After successful authentication, Amazon Cognito returns user pool tokens(Three tokens) to your app. You can use the tokens to grant your users access to your own server-side resources, or to the Amazon API Gateway. Or, you can exchange them for temporary AWS credentials to access other AWS services.
The three tokens are ID Token(JWT), Access Token, Refresh Token. The refresh token can be used to retrieve new ID and access tokens. Once you login to a mobile app, you are not needed to log in each time when you close and open the application and this functionality is implemented using refresh tokens.
https://docs.aws.amazon.com/cognito/latest/developerguide/amazon-cognito-user-pools-using-tokens-with-identity-providers.html
What about Amazon Cognito hosted UI ?
App clients can be configured to use built-in Cognito webpages for signing up and signing in users. When using the hosted UI you can enable both the Authorization code grant and the Implicit code grant, and then use each grant as needed.
https://docs.aws.amazon.com/cognito/latest/developerguide/cognito-user-pools-app-idp-settings.html