Yes, you can send and receive SMS texts on your Mac - although you'll need to link the Mac to an iPhone.
Apple brought text-style messaging to the Mac a few years back with Messages (sent from the Messages app in Mac OS X). With Messages you have always been able to send iMessages from a Mac to other people using an Apple device.
Before the launch of OS X Yosemite in 2014, if the other person wasn't using an Apple device then you'd be stuck. You'd have to go back to the iPhone, which can send and receive regular text messages.
With Yosemite, and its new Continuity features, all that changed. And El Capitancontinues to offer the ability - if you have an iPhone, and a Mac - to use Messages to send and receive text messages.
How does Messages for the Mac send texts?
In most respects the Messages app in Yosemite and later is like earlier version of Messages, which in turn is like most messaging apps. What Yosemite added was the ability to send and receive text messages. This shouldn't be possible because the Mac doesn't have a SIM card, and isn't a phone. But if you have an iPhone and a Mac, OS X links the two together.
If your Mac and iPhone are using the same Apple ID, the devices know when they are on the same network together and Messages syncs all the text messages from the iPhone. So you get to see all the text messages alongside all your regular iMessages.
If you send a message from the Messages app on a Mac it first tries to send it as an iMessage; but if the person hasn't got iMessages enabled (or isn't using an Apple device) it sends it as a standard text message. OS X doesn't actually send the message itself: it directs the message via the iPhone to the person, and the message is synced on your iPhone and Mac.
The upshot of this is that you can send and receive text messages with anybody directly using Messages on your Mac. Once this is set up it's a great way to use Messages.
Set up text message forwarding
First of all, make sure that you've installed Mac OS X 10.10 Yosemite or later - we'd recommend using the latest version, El Capitan. (Here's how to update your Mac to El Capitan.) You'll also want to install the latest version of iOS (iOS 9.3.2 at time of writing) on your iPhone.
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