There really isn't any doubt as to the proper translation of the neuter-gendered noun σῶμα (sōma), which is "body."
The word ψυχικόν (psychikon) is the adjectival form of the noun ψυχή (psychē).
Likewise, the word πνευματικόν (pneumatikon) is the adjectival form of the noun πνεῦμα (pneuma).
ψυχ- (psych-) is the actual root of the word ψυχή. Essentially, the suffix -ικόν is added to the root to form the adjective. Hence, ψυχ- + -ικόν = ψυχικόν. Likewise, πνεῦματ- (pneumat-) is the actual root of πνεῦμα. πνεῦματ- + -ικόν = πνευματικόν. The suffix -ικόν changes the noun into an adjective.
The Greek word πνεῦμα is almost always translated into English as "spirit" in the KJV (for example). Thus, "spirit" (πνεῦμα) becomes "spiritual" (πνευματικόν).
The problem is that the Greek word ψυχή is translated several ways in the KJV. Of its 105 occurrences, it is translated as "soul" 58 times and "life" 40 times. As you can imagine, this makes determining the meaning of ψυχικόν a bit difficult.
In his Vulgate, Jerome translated into Latin as follows:
It seems Jerome what quite consistent in his translation of πνευματικόν since he always translates it by a declension of the adjective animalis. The adjective animalis essentially refers to something that is living and breathing. A related noun, anima, essentially means "breath." Yet, another noun, animus, basically was understood to be the "soul." Well, frankly, there's no real way for us to understand how Jerome is using animalis unless you read his commentary on these particular books of the NT, but unfortunately there aren't any.
The Differences between the σῶμα ψυχικόν and σῶμα πνευματικόν
If you want to understand the difference between the σῶμα ψυχικόν vs. the σῶμα πνευματικόν, there are two methods.
First, you can read the scripture that describes Jesus' acts after he was resurrected from the dead, since Jesus possessed the spiritual body after he was resurrected. We know he possessed flesh and bone (Luke 24:39). He could vanish instantaneously (Luke 24:31). He could walk through obstacles unimpeded (John 20:19, 20:26). Scripture may describe a few other details, but nothing as extraordinary as these.
Next, you can read Paulos' own description in 1 Corinthians 15, since he describes the differences.
The σῶμα ψυχικόν is what we currently possess. It is "corruptible" (φθαρτὸν; 1 Cor. 15:53), meaning it decays or corrupts and requires sustinence to live, including oxygen and blood, nutrients, and water. It is also mortal (θνητὸν; ibid.), meaning it is subject to death.
Man does not simply "swap" a corruptible body for an incorruptible body upon his resurrection from the dead. Rather, the corruptible body is changed into an incorruptible body. Meaning, Jesus still possesses the same body before and after his resurrection from the dead. However, the body he had before his resurrection endured a change.
This change occurs in one of two ways: either man dies and the body is changed upon his resurrection from the dead (1 Cor. 15:36) (this happened to Jesus). Or, God will cause the change in the man's body while the man is still living (1 Cor. 15:51). This change is instaneous (ἐν ἀτόμῳ; 1 Cor. 15:52).
The σῶμα πνευματικόν is what we shall possess upon the occurrence of the resurrection of the dead. (Again, the change occurs at the resurrection of the dead. Some, however, will be changed while they are still living at the time of the resurrection of the dead, while those who have actually died will be resurrected and changed instanteously at the time of the resurrection of the dead.) This body is incorruptible (ἄφθαρτος; 1 Cor. 15:52), meaning it does not decay or corrupt. It does not require sustinence; therefore, it does not need oxygen, water, nutrients and food, and perhaps, even blood.
It is also immortal (ἀθάνατος; 1 Cor. 15:53), meaning it cannot die. Now, a caveat is this: nothing is inherently immortal but God Himself (1 Tim. 6:16). The reason that the immortal body is immortal is because God wills it to be immortal, giving it eternal life (John 10:28). (It is obvious that God can destroy anything if He so desires.)
Because the meaning of ψυχικόν is so vague, I think we can learn more about the meaning of σῶμα ψυχικόν based on Paulos' description of it, rather than having to guess at the actual meaning of the word.