Let the waters under the heavens be gathered into one place, and let the dry land appear. - Genesis 1:9

I was wondering how you can tell the difference when the Bible speaks about physical death or spiritual death. In the English translation it just says death so I didn’t know if the Hebrew or Greek makes it clear. Specifically I’m wondering about these two verses. (Gen. 2:17, I Cor. 15:26)

This is a very good question.  In Genesis 2:17, God told Adam that if he ate of the fruit of the tree of the knowledge of good and evil, “he would surely die.”  But what kind of death was God referring to, spiritual or physical? 
To answer this question, let us go to the next chapter and see what happened when Adam and Eve rebelled against God’s command.  In third chapter of Genesis two things happened: 1) in verse 7, Adam and Eve “knew they were naked” and 2) in verse 19, they were banished from the garden, “unto dust thou shalt return.”
What we see here is two types of death, one spiritual and the other physical.  Both are ultimately inseparable.  For Adam and Eve, they were not created spiritually dead nor were they going to die physically.  Death is a not a natural part of a perfectly created world, death is judgment; death is a punishment for sin.
So, the death mentioned in Genesis 2, is further explained in chapter 3 as being both spiritual and physical deaths.  We know the separation we have from God, before salvation, is a total separation. 
Further, consider this, what aspect of our physical life does not have spiritual implications?  When we speak a hateful word, it comes from a heart of anger.  When we take something that isn’t ours, we have coveted something someone else has; this is a desire born of a selfish heart.  Can we separate the implications of a physical death from a spiritual death?
In the 3rd chapter of John, Nicodemus asks Jesus, “How can a man be born when he is old? Can he enter the second time into his mother’s womb, and be born?”  Jesus responds by saying, “Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.  That which is born of the flesh is flesh; and that which is born of the Spirit is spirit.  Marvel not that I said unto thee, Ye must be born again.  The wind bloweth where it listeth, and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit.”
Since Adam, we are born of water, that is to say, we are born physically from our mother, but there is a second birth required because when we are physically born, we are born spiritually dead and separated from God.  A second birth, a birth of the spirit is necessary for a complete fellowship with God.  This is where we hear the phrase, “being born again.” 
Death is a natural part of life, but only from the perspective that we live in a fallen world.  Not from the perspective that God created Adam and Eve perfect and without sin.  This standard of perfection will be attained one day when Christians, born again spiritually, receive their new bodies.  A picture of this perfection without death is seen in Revelation 21:4, “And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.”


As far as the 1 Corinthians 15:26 passage, your question is about which type of death is being referred to here.  As with any passage, context is key.  So, what is our context here?  If you look back at verse 20 we see Paul talking about the physical resurrection of Jesus.  Jesus is the “firstfruit of them that slept.”  Jesus is the first of the resurrection of which all who are in Christ will follow.  But we must not miss what Paul does here in the next verse; he reminds us of Adam and the impact Adam’s choice had upon humanity, “For as in Adam all die.”  Then, stepping forward to verse 26, “the last enemy to be destroyed is death.”  Your question, which death?  Jesus will defeat the totality of death, both physical and spiritual.  There will be no more death one day; no more people being born dead with a fallen sin nature and no more people being subject to the impact of physical death upon humanity.  My answer is: both.

To try and conclude things, there are times when we read about the death of someone and we can clearly see that this is referring to a physical death.  Example: When Stephen was stoned to death in Acts, we know that this was a physical death. 
But when we look at passages like Genesis 2-3 and 1 Corinthians 15, we can see that more is going on.  I am a born again Christian, meaning spiritually I am born again, but I have not yet received my new physical body.