Note: There's not really an activity associated with this information, although if you want to experiment with direct messaging, by all means do so! I'll need to follow you if you want to direct message me as an experiment, so tweet me (@OnlineCrsLady) first so that I can follow you, and then we can then try a DM just for practice.
Replies. Earlier, you had a chance to see how Twitter conversations work by replying to a tweet. By definition, a Twitter reply means that someone's @handle is the very first thing that appears in the tweet. If someone's @handle appears elsewhere in the tweet, then that is a "mention," not a reply. Both replies and mentions generate notifications to the person, but only replies build into a conversation.
Replies and Conversations. To see how a Twitter conversation works (and it's admittedly kind of awkward), remember that you can click on the day-time stamp of any tweet to see it on a page of its own; if it is part of a conversation, the other tweets in the conversation will appear on the page with it. Twitter also sometimes provides a "View Conversation" link that pops open the conversation inside the current page without taking you to a separate page. So, if I see this retweet, I can click on the time-day stamp:
So, as you can see, it's all a little complicated. Twitter is sort of a place for conversations, but it's not exactly built around the idea of conversations. Instead, it is built around people... and those people might be engaged in conversations, or not, based on whether or not they are replying or being replied to. You can join in on a conversation and also see what others are saying, but it might take you a couple of clicks to do that!
Direct Messages. And that brings us around to direct messages. If you do want to have a private conversation at Twitter, you can use the Direct Message feature to do that. I personally don't use this feature very much, but you can find out all the basics at the Twitter help page on Direct Messages. You will find detailed information on that page about sending and receiving direct messages, along with FAQs. So, if you are looking for a private channel to use at Twitter, in addition to the public space, then you can consult that page for all the information you need to do that.
Protected Tweets. It is also possible to make a "private" Twitter account, although I'd recommend against doing that. Twitter is very much designed for public users rather than private users; if you don't have any interest in participating in that public space, Twitter is probably not a good tool for you. If you want to find out more about protected tweets and private Twitter, here is that Twitter Help page: About Public and Protected Tweets.
NEXT STEP => Now that you've learned about private conversations via direct messaging, it's time to learn about some privacy and security features at Twitter: muting and blocking.