Sometimes a carotid artery can narrow and be 'asymptomatic', meaning that the patient has no symptoms. Symptoms do vary from patient to patient.
When symptoms do appear, they are usually the same symptoms as a stroke or TIA (transient stroke), including weakness or numbness on one side of the body, trouble with coordination, facial drooping, trouble with speech (slurred speech or trouble speaking/understanding language) or cognition, and at times vision loss. Rarely patients can have limb shaking in association with their stroke/TIA symptoms.
If you have any of the symptoms above, you should seek immediate medical attention or head to the Emergency Room.