Manifestation is essentially a process of transformation, becoming that which you seek, and actively calling in the life you desire. It's similar to, and directly works with, the law of attraction, which we'll get into shortly.
Manifesting is all about pairing mindset with action to get real results, and as psychiatrist Anna Yusim, M.D., previously explained to mbg, it was popularized by Rhonda Byrne's book and movie, The Secret, describing the universal principle that our thoughts create our reality. "We are what we think, consciously and unconsciously. In other words, what we attract into our lives reflects the contents of our mind," Yusim explains.
Think of it this way: If you want to build a table, you start to visualize in your mind the process of building that table. You know in your heart that you can build the table with the right tools and supplies. You start to get a clear idea of what you need to do in order to build that table. Then, you get the supplies you need and you build it. Boom: table, manifested. The key is the thought came first, you held onto it, and you manifested the table into reality.
While that's a more concrete, tangible example, that is the essence of manifestation, just replace the table with "love," or "abundance," for example.