Turn iMessage off and back on, or turn iMessage off to see if texts can be sent and received.
Make sure the contact is not on your block list. To do so, go to Settings > Phone > Blocked, and confirm that the user is not on any block list.
If you have an old iPhone or another iOS device that you sometimes use for iMessages, try toggling iMessages off there and sign out of your Apple account. You can sign out by going to Settings > Messages > Send & Receive and tapping on your Apple ID, then choosing Sign Out.
Delete your current conversation with the problem contact(s) and start a new one.
Delete and re-enter said contact(s).
Simultaneously hold the Sleep/Wake and Home buttons to reset your device.
Try going to Settings > General > Reset > Reset Network Settings. You will have to re-enter some passwords and set network connections up again after this.
You could try a factory reset of your iPhone.
Contact Apple Support, or take your device to an Apple Store to get it checked out.