You can also press "Delete All" in the top left of a message thread to erase texts.

When you send messages to, or receive messages from, someone on your iPhone, iOS 7 condenses the conversation into a single thread. If you no longer want the Messages app to display a conversation from a contact, or if you don't want anyone to see a specific text you've sent or received, you can delete the messages from your device.

Removing Threads or Single Messages

Open "Messages" from the Home screen and then select "Edit" from the top left. Press the "-" icon next to the message thread you want to remove and then press "Delete" to confirm. When you're finished, press "Done." To remove a specific message without deleting an entire thread, select the appropriate phone number or contact from the list, and then press and hold the message you want to delete. Touch "More," select the trashcan at the bottom left and then press "Delete Message" to confirm.

Removing Multiple Messages

You can also delete multiple texts at a time from within a single thread. Follow the same process you would to remove a single message, but before you touch the trashcan, scroll up or down through the thread and touch the circle next to each message you want to remove. Touch the trashcan and then press "Delete n Messages," "n" representing the number of texts to remove.