Quick answer: Your Matches and conversations are on the Chat tab. Members who Liked you and Swipe Notes you've received are on the Likes tab.
- Chat: every Match and conversation. Matches don't expire.
- Likes: members who Liked you (First Class), plus Swipe Notes you've received.
- To end a Match and its chat, use Remove Match.
Chat
When you and another member Like each other, it's a Match, and it appears on your Chat page. Tap a Match to open the conversation. Chatting with Matches is free, and your Matches and conversations are kept, so they don't expire. First Class members also get Read Receipts, so they know when their messages have been read.
Likes
The Likes tab shows members who Liked you and Swipe Notes you've received. See How does the Likes You tab work?
Removing a Match
To end a Match, open it and select Remove Match. This also removes your conversation, and it can't be undone.