“And it shall be in the last days, says God, I will pour out of my Spirit on all flesh, and your sons and your daughters will prophesy, and your young men will see visions, and your old men will dream dreams. Yes, and on my servants and on my handmaidens in those days will I pour forth of my Spirit, and they will prophesy.”
Joel 2:28 in LXX reads, ‘And it shall come about afterward, that I will pour out my Spirit on all flesh'. Peter paraphrases ‘afterward' as ‘in the last days' (or quotes from a collection of sayings which has done the same). Joel's prophecy does in fact have reference to the last days and stresses that it is dealing with ‘the day of Yahweh' (Joel 1:15), the time when God chooses to work among men. However, ‘the last days' is a significant phrase for it is the phrase used in Isaiah 2:2-4; Micah 4:1 referring to the time when God's Temple would be miraculously exalted, when the peoples would respond to Him, and when His truth would go out to the world. Nothing could be apter for the Day of Pentecost. So Peter is linking Joel 2:28 with Isaiah 2:2-4. The coming day of Yahweh is also often called ‘that day' (Isaiah 2:11; Isaiah 2:17; Isaiah 4:2; Isaiah 11:10 etc.), that is, the one coming at the end. So Peter is simply by his changes putting the quotation in its true context. The quotation is otherwise similar to both LXX and MT except for the reversal of the words with regard to young and old men and the final addition of ‘and they will prophesy' (which is merely repeating what has been said for emphasis. This is probably a preacher's use of a text where he is stressing the salient points).
It should be remembered that to the Apostles the days which had now begun were ‘the last days. See Hebrews 1:2; Heb 9:26-28; 1 Corinthians 10:11; 2 Timothy 3:1; 1Pe 1:20; 1 Peter 4:7; 2Pe 3:3; 1 John 2:18. The last days may have lasted two thousand years, but to God that is but a short period in the night, ready for the coming of the Day..
The prophecy promises ‘the pouring out of the Spirit on all flesh'. In context ‘all flesh' means all types of people, sons and daughters, male and female, young men and old men, menservants and maidservants. It does not necessarily include non-Jews (compare Ezekiel 21:4-5 which is addressed to Israel and where ‘all flesh' can only mean Israel). Thus the promise, which he is now declaring as in process of fulfilment, refers to a general and all encompassing pouring out of the Spirit on all God's people.
And that day was also to be marked by ‘prophecies' such as these they were hearing in their own languages from these men about whom they were commenting. It was especially to be a time of prophesying, and also one of visions and dreams (which will come out later in Acts). This identified what was happening with Joel's prophecy. It was all evidence that the Holy Spirit, God's own Spirit, was being poured forth.