A place to keep your horse. If you are lucky this is your own property. Make sure you are allowed horses in your area and have sufficient land for a horse. A barn is not required, but a shelter is. The property should be well fenced and have a water supply. You can pay a stable to board your horse, usually there are three options; pasture board where your horse is outside all the time, tie-stall board, where your horse is out in the day and in a narrow standing stall at night, and box-stall board, where your horse as a slightly roomier stall, you also pay more.

Money

Like most luxuries, horses come with a price tag, many people forget the ongoing costs incurred by owning a horse, and the emergency expenses. When planning a budget to see if you can afford a horse, make sure you over estimate emergency expenses, ideally you should start a savings account for such matters.

Feed and Water

If you are boarding your horse at a stable, these are included in your fees. If not you will need to buy your own. While some horses do well on pasture, others need supplemental feedings of hay, grain, or more. For water, you will need a bucket, water trough, or automatic watering system.

This is my photo of a mini mare, all I really need for her, are the basics.   

Halter and Lead

When you buy a horse it typically comes with a halter and lead shank, but may not. These are two things a person really should have because a halter broke horse is much easier to work with and for emergency care, and farrier work it is a requirement.

Hoof Pick

Believe it or not, many people have horses and pretty much ignore them, but picking their hooves is a very simple task that keeps their feet in better condition, free of rocks and manure.

Image via Wikipedia

Brushes

There are a slew of brushes a person can have to use on their equine. You will want a Dandy Brush, a Curry Comb, and a Mane and Tail Comb. Any other brushes are extras, such as tools for braiding their manes, and shedding them out.

Salt and Mineral Blocks

If you are keeping your horse at a stable they should have these already, if you are keeping your horse at your home, you will need to get these.

Riding Helmet

If you are a new rider, have a green horse, or will be jumping, a helmet is a great purchase.

Image via Wikipedia, showing a horse who does not wear a bridle with a bit, but rather uses a Bosal.

Bridle

Sometimes the owner of the horse will sell the one they have used to you. Different shaped and sizes of heads will determine what size and type of bridle your horse will need, so buy the horse first, bridle second.

Bit

The bit is the part that goes in the horses mouth, there are a huge range of bits from gentle to severe, you cannot just buy one bit and assume it will work on any horse. Some horses do not even use bits, so again, buy the horse first, then the bit.

Saddle

You might be okay to buy a saddle before you buy your horse, but may find the saddle does not fit the horse properly and will cause painful rubs, so again, horse first, saddle second.

Girth or Cinch

Fat horse or skinny horse? This will determine what size of girth or cinch you will require for your equine.

Saddle Pad

This can really be purchased at any time, but it might be easier to purchase it at the same time you buy your saddle.

Other Tack Items

Martingales, tie-downs, breast collars. Depending what style of riding you are doing, you may need other items.

Leg and Hoof Protection

As required by your riding style, a horse may need bell boots, leg wraps, splint boots, etc.

Wikipedia

Image via

Extras

• Treats. You can buy treats earlier, but they would go stale. Treats can be carrots, sliced apples, or store bought horse cookies.
• Saddle Rack or Cover.  These will keep your saddle looking better than if you keep it on the floor.
• Saddle Soap and Tack Cleaning Supplies.  Again, these will keep your saddlery in good condition. 
• Ridding Accessories. Whips, spurs, chaps. Buy them if needed.
• Buckets and Pails. These are always handy to have for carrying treats and grooming supplies.
• Bathing Supplies. Horse shampoo, conditioner, sponges, and a scraper (to remove water, it also works for removing sweat too)
• Blankets. For winter time, if a horse is stabled at night and you live in a cold climate, the horse will not grow a warm coat for protection outside, as such providing a blanket is beneficial.
• Vet Rap. This is good to have for emergencies, and can be used to wrap a horses tail for trailering.
• Medical Supplies. It is good to have a few items for emergency care, an antiseptic is important to have, along with some clean towels.
• Bug Spray. Not every horse is bothered by bugs but for those who are, bug spray might be handy.
• Clippers. Sure you can borrow some, or use scissors to keep your horses bridle path trimmed, but owning your own clippers is a good idea too.

Image via Wikipedia

People

You will need to have a reliable farrier and veterinarian. The farrier is the one who trims your horses hooves and applies shoes if needed. The veterinarian is the one who will give your horse its required shots, tend to medical emergencies, and deal with any dental problems. When selecting a farrier and veterinarian, get recommendations from friends and fellow horse people.

You may or may not need a trainer or riding coach. Make sure you find one that is willing to work with you and the horse as a team, and is familiar with the breed of your horse.