This is probably the most controversial verse in the Bible. That is because it witness against certain doctrines.
Note that verse 36 is a question, and verse 37 is the answer, so without it verse 36 does not make sense.
Also note that some bibles even quote verse 37 different(NAV:1983 trans.) in the footnote to make the answer less contraversial.
Many discredit this verse because it is not present in all the manuscripts, but that could also mean that it was not included because it went against doctrine.
It is important to note that this verse was referred to by early church fathers.
Early church fathers who witness to it's being a part of inspired Scripture and quoted or referred to Acts 8:37 are Irenaeus, Cyprian, Chromatius, Tertullian, Ambrosiaster, Pacian 310-391 A.D., Ambrose 340-397 A.D., Augustine and Theophylact.
Many church fathers who lived before anything we have in the way of Greek copies directly quote this verse, including Irenaeus 178 A.D., Tertullian 220, Cyprian died in 258, as well as Ambrosiaster 384, Ambrose 397, Augustine 430, and Venerable Bede of England in 735.
For example, Cyprian (200-258 A.D.) supports the inclusion of verse 36-37 Textus Receptus when he says, "In the Acts of the Apostles Treatise 12:3: Lo, here is water; what is there which hinders me from being baptized? Then said Phillip, If thou believest with all thine heart thou mayest." (The Treatises of Cyprian )
Irenaeus (115-202 AD), Against Heresies 3.12: "Philip declared that this was Jesus, and that the Scripture was fulfilled in Him; as did also the believing eunuch himself: and, immediately requesting to be baptized, he said, I believe Jesus Christ to be the Son of God."
Augustine (354-430 AD), Sermon 49: "The eunuch believed on Christ, and said when they came unto a certain water, See water, who doth hinder me to be baptized? Philip said to him, Dost thou believe on Jesus Christ? He answered, I believe that Jesus Christ is the Son of God. Immediately he went down with him into the water."