The real-time constantly updating nature of Twitter means that a tweet you send might be missed by your followers if they log in ten or twenty minutes after it was sent. How long the tweet remains on their timelines depends on the number of other tweets published in the meantime. Every tweet you send is stored permanently on your profile and can be downloaded in your Twitter archive.
Tweets in the timeline
Log in to Twitter and you'll be met with the timeline -- a chronological list of tweets from the people you're following, with the newest at the top. As more tweets come in, older ones are pushed out of view (though you can still scroll down to see them). How long a tweet lasts on your timeline (how long it stays on screen for) depends on the number of accounts you're following and how busy they are.
Tweets on profiles
All published tweets are stored permanently on profile pages, where you can also view replies and retweets unless the account is protected. If you want to review the tweets from a particular account over a longer period than the timeline allows, visiting the relevant profile page is one way to do it. Twitter profiles make use of an infinite scroll feature so you can keep going back through time to find older tweets. There is no expiry date for tweets stored on a profile.
Tweets in search
Twitter's search feature enables you to look for keywords or hashtags in older tweets once they've disappeared from the timeline (or if they weren't on the timeline in the first place). How long a tweet is available in search depends on the number of matches your query returns -- if the search matches a lot of tweets, you might only see updates from the last few minutes. Twitter does include older tweets in its search results where relevant and automatically returns the most popular matches based on retweets and clicks.
The Twitter archive
Tweets are stored permanently on each user's profile and as an extension of this you can download your own Twitter archive in its entirety. Follow the "Request your archive" link from the account settings page and you'll receive an email when your archive is ready to download (which could take a day or two). This feature means none of your tweets are ever lost and can be stored for as long as you like.