If you are physically active and can run about 3 miles in one stretch, you can train to run a half-marathon in about 3 months. Please note that your objective is to finish the race around 2 hours, mostly running.
If you read the vast literature on running, you will find many methodologies to prepare you for a long distance race, and at times they might be confusing. The main goal of all those training plans is the same though: prepare you to run longer distances gradually to avoid training injuries.
This guideline is largely based on how I got ready for my first 1/2 marathon. The focus here is not to improve your time; but it is to make you run long distances smoothly.
1. Start running at least 3 days a week and get a point where you can run at least 3 miles in one stretch; some walking is ok, but mainly running; don't worry about speed at this point.
2. Increase the distance you cover in one session by approximately 10% once you are comfortable doing a specific distance. The steps could be 3,4,5,6. Make sure that you try the next longer distance only after you can run the current target distance comfortably. Time required in reaching 6 miles in one stretch would depend on your ability to run, but make sure you don't rush and only reach there gradually to avoid training injuries.
3. After you can start running 6 miles, you can do long runs on weekends. In this schedule, you will run short distances between 3 to 6 miles (longer the better) 3 days a week and a long run on the weekend which will be increased by approximately 10% every week, like 7,8,9,10,11,12 miles. Run a short distance like 3 miles before the day for long run and take rest the day after long run.
4. 1/2 marathon is 13.1 miles, but once you run 12 miles couple of times, you are ready to run the race.Your training should be wound down at least 2 weeks before the race. During this period, don't stop running, but run at least 3 days a week logging 3 to 5 miles in every run.
5. Do weekly hiking as a cross-training to build strength, esp. during the first part of the training before you start the weekly long runs. This is optional but it will be very helpful if you haven't been very active prior to starting this training.
Based on this methodology, for someone who hasn't run much before, the training schedule would look like this, if you make 10% progress in distance covered every week:
Week 1: 3,3,3,Hike -- 9M
Week 2: 3,3,4,Hike -- 10M
Week 3: 4,3,5,Hike -- 12M
Week 4: 4,4,6,Hike -- 14M
Week 5: 3,4,3,7 -- 17M
Week 6: 4,5,3,8 -- 20M
Week 7: 4,5,3,9 -- 21M
Week 8: 4,5,3,10 -- 22M
Week 9: 4,5,3,11 -- 23M
Week 10: 4,5,3,12 -- 24M
Week 11: 4,5,3,12 -- 24M
Week 12: 3,4,5,4 -- 16M
Week 13: 3,3,2,13.1 - RACE WEEK
The schedule can be compressed if you are capable of skipping the early training weeks and start later in the schedule.