stslavik in an answer related to Kenjutsu mentions the concepts of mushin, fudoshin, and zanshin as part of Fudo Myo-o. While I am somewhat familiar with all three, I am seeking a more in depth explanation as to how they relate to kenjutsu and aikido.
You're asking a lot of interconnecting but separate questions. Depending upon the view and the school, you will generally get different answers, but in Mikkyo (Esoteric Buddhism), these concepts (Fudōshin, Mushin, and Zanshin) are all derived from the teachings of Fudō Myō-ō. There is an order to these things, as I was taught it, so I will answer in that [relatively descending] order, attempting to keep it as simple as possible.
Who Is Fudō Myō-ō?
Fudō Myō-ō is the Japanese name of Acala, one of the Five Wisdom Kings of the Womb Realm. His name in Japanese roughly translates to "Immovable Wisdom King", is the patron of the Yamabushi, and, according to Shingon Buddhism, is not technically a Buddha. Fudō Myō-ō is the destroyer of delusion and a principal protector of Buddhism, and is the "Immovable Wisdom King" for his mind being forever unmoved. He carries a sword in his right hand, to cut away delusions and ignorance, and a rope in his left hand to bind violent and uncontrolled passions or "evil forces").
What is Fudōshin?
Fudōshin is the immovable mind; calm, serene, perfectly stable, and unchanged. It is a state of equanimity, and imperturbability and acceptance of the present moment. It can be viewed that both Mushin and Zanshin are aspects of Fudōshin.
What is Mushin?
Mushin is the state of no mind, or the mind unfettered by ancillary or superfluous thoughts. It is the mind in the moment, clear and unobstructed.
What is Zanshin?
Zanshin is the state of the remaining mind; the mind maintained in the moment, relaxed but alert. While many martial arts attempt to "grade" zanshin by looking for the continued ready stance, this is not indicative of the minds awareness. Instead, the mind in zanshin has overcome an impediment and is ready for another, while still remaining in the present moment. This is sometimes called "relaxed awareness".
What is the Mind?
I've purposely left this for last in the discussion of Fudō Myō-ō: the mind referred to above is not the mind (in a philosophical, cogito ergo sum sense) necessarily as we think of it, but rather the Buddha Nature – the capacity for sentient beings to become Buddhas.
The relationship to Kenjutsu and Aikido
It is incorrect to say that these concepts relate to kenjutsu. Kenjutsu is a broad category, comprised of various different arts (For example, Kukishin-ryu kenpo, Niten-Ichi-ryu Nito Seiho, and Yagyu Shinkage-ryu Kenjutsu are all individual styles of kenjutsu with different philosophies). Schools of kenjutsu with ties to Buddhism may teach some or all of the aforementioned virtues, each with different takes on the importance. Further, it is not necessary that it be a sword art for them to be viable (Fudōshin, for example, is taught as an important aspect of the taijutsu of Shinden Fudō-ryu). Fudō Myō-ō is commonly revered, however, in Japanese martial arts for the stillness of his mind.
Aikido, being derived from training in Daitō-ryu Aikijujutsu which in turn likely developed a great deal of its taijutsu ("body skills", which includes positioning [taisabaki]) from the taijutsu of Kashima Shinden Jikishinkage-ryū Kenjutsu and Ono-ha Ittō-ryu Kenjutsu, likely took a great deal of the Buddhist teachings popular among Samurai of its ancestor schools. When I studied aikido in an aikikai school, I never heard more than mushin ever talked about.