Everyone wants that special connection of trust and affection with their horse that takes them from money-sucking-dependent-animal-that-is-a-lot-of-work to best-friend-on-the-entire-planet-I'd-do-anything-for. The forming of that bond is different for every person and horse–and it usually takes time and patience–but there are some simple ways to make sure you're initiating a relationship in a meaningful way to them.
Some horses seem to intuitively know you're their human, and from day one they look to you as their leader and companion. These special horses seek human connection sometimes even more than connections with other horses. The owner of a horse with this personality type needs only to show up to spend time with their horse occasionally, feed a few treats on occasion, and they'll have their buddy in the palm of their hand.
For most of us, though, we have to work a little harder to win our horse's trust and affection. First, observe our horses closely whenever you interact with it to find out what makes them relaxed, content, excited, nervous, upset, etc. Most horses have evident likes and dislikes that are personal to them. For example, my opinionated pony loves having his butt and face scratched and groomed, but dances around and swishes his tail when I groom his belly. I still groom his belly when needed, but I do so briefly and gently out of respect for the fact that it is sensitive, and I spend extra time on the areas I can see he enjoys.
Be careful, however, to not completely avoid doing things your horse gives you push-back on. For instance your horse might not like baths, but he/she does need to learn to endure them once in a while without putting up a huge stink. Teaching them to stand still for a bath isn't going to make your horse hate you, it will actually make them respect you more. Above all else, your horse wants to know that someone is in charge, keeping them safe from threats, so that they can relax and live life in peace. Being a strong herd leader means being affectionate when appropriate, fair and considerate, but also knowing when to put your foot down and demand respect.
If your horse is constantly challenging your role as leader, you may feel like you're nothing but a disciplinarian to them. Stick with it, and be quick to recognize and reward the peaceful moments when your horse has accepted your role as Boss. They will get the idea pretty quickly if you are consistent. Consistency really is the key.
If your horse is a true follower and doesn't challenge your position, you still have to be a strong leader for them or they will likely act skittish and insecure around you, sensing that there isn't a strong leader present. Doing some ground-work exercises where you ask your horse to move away from you (you step towards their hind end and they swing their butt away from you, you step towards their shoulder and they side-step their shoulder away from you, you step towards their head straight-on and they back up) will show them you're in charge of them and, because those exercises mimic the way horses establish rank in their own herd, it will eventually relax them. When your nervous horse becomes relaxed (lowering of head, chewing, deep sighs, soft eyes and ears), reward them.
Speaking of reward, treats can be a great secret weapon to unlocking your horse's affections. I do hesitate a little bit to talk about using treats, however, because they can create a real monster if you aren't extremely careful with how you use them. First, it's really important that you understand your horse's base personality. Is he pushy and naturally inclined to want to be the leader? Proceed with caution. Treats at the wrong moment will reinforce unwanted behavior, so ONLY give them as a clear reward for a good ride or an excellent response to something you asked for, like loading into a trailer.
Personally I use treats to catch my pony, because he can be hard to catch in the pasture. After several instances of always having a tasty morsel for him with me, he no longer runs away from me and willingly stands still while I put the halter on. He has never nipped or acted pushy with me regarding this ritual of a treat in exchange for his good manners during catching, but if he did I would stop and come up with a new solution.
There is a big disadvantage when you board your horse at a farm where someone else feeds them every day: You don't get to have the identity of Wonderful Amazing Super Duper Food Person to your horse. And if you come out twice a week solely to ride your horse, your identity to them is probably something like Makes Me Work Hard Person (this happened with my pony and me, hence why he became hard to catch.) There is a way around this, and that is to spend time with your horse NOT working. Groom them if they enjoy being groomed, let them loose in the arena and play games like Follow Me, take them for a short walk to graze on some new grass, or do light, fun and stimulating work like trail-riding. Consistency is really key here, because horses thrive on a solid schedule. It helps them feel secure with their world if they can count on you showing up every other day at roughly the same time to do something with them, whether that's work, play, or just grooming.
Trail-riding is a great bonding experience because your horse is away from its comfort zone, is encountering new things, and has to rely on you to tell him/her where to go and whether or not that bicycle that is approaching is something scary or mundane. If your horse has never trail ridden before, it's important you go out for the first time with a couple other horses who are seasoned at it and very calm, and that you are confident and comfortable enough to not get tense or scared if your horse gets nervous. Over time, this will hopefully become an activity you and your horse really enjoy and that promotes bonding.
Every horse is different, but consistency and patience is key, as is working on your ability to be a confident and reliable leader for your horse. The better you establish yourself in that role, the more peaceful and relaxed your equine friend will be, and you can think about the dynamic less and enjoy each others company more. Have fun with the process!