Echofon for Twitter is a program that syncs unread Tweets across all of your devices, like iPhone, tablet or Android mobile. This application helps you check the newest Twitter posts from the users you follow, send private messages, post picture or links, and much more.
After you log in to your personal Twitter account from the main window of Echofon, you can begin reading the latest Tweets, find Twitter users, and start following other people's feeds. I like the fact that the program supports account switching (you can use multiple accounts).
In order to sync the unread Tweets across your mobile phone or tablet you just go to Preferences menu and click the Sync button (you can always stop sync from the same window). This way, when you change from Mac to other device, you read the Tweets posted after your last sync.
This program also features an integrated search tool that helps you find users and Tweets within moments.
However, if you don't mind reading the same Tweets twice, I'd say it wouldn't worth installing an additional program on your Mac and you could just use Twitter app for Mac instead (which brings more features and is free of charge).