You are familiar with the root of the word "m'kayeim" ("u" is the prefix meaning "and") from its usage in the most familiar of blessings – "she'he'che'yanu V'KI'Y'MANUI v'higianu..." – and from the liturgy surround "mi khamokha" – "tzur yisrael KUMA b'ezrat yisrael." But when it appears in the Amidah the three meanings of "l'kayeim" are all on display.
The first meaning of the word is "establish." To speak of God as m'kayeim is to ascribe the power to make something fixed or permanent. One of the most intriguing concepts in Jewish law is the "bar kayama," the notion that an infant must survive thirty days to be consider a fixed rather than a fluid life. God's participation in an action enables it to become more than transient.
The second meaning is "sustain." In this sense, something already in existence continues due to the nurture and support provided by God. In the blessing mentioned above, God is the one who keeps us in life AND SUSTAINS US in that life.
The third meaning of m'kayeim is "fulfill." The word is used not in the context of the feeling of accomplishment that follows the completion of a task, but the literal meaning of the English word, "fill full," that is, to maximize capacity and/or potential.
The word itself is very powerful in this grammatical form, the causative. In its simple form it means to arise, stand, or get up. At first reading, those words do not seem so infused with power, yet "to take a stand" is one of the most inspirational phrases in our language. The ability to convey that authority and courage is that much more impressive.
What exactly is it that God establishes, sustains and fulfills? That's the next question.