The question of eating blood goes right back to God’s instruction to Noah after the flood.
Every moving thing that lives shall be food for you. And as I gave you the green plants, I give you everything. But you shall not eat flesh with its life, that is, its blood. (Genesis 9:3-4)
This was repeated in the law given through Moses (e.g., Leviticus 17:10-14; Deuteronomy 12:16,23).
The Jerusalem Council
In the New Testament, the early Christians had to decide what parts of the Israelite law would be required of non-Jews. Guided by the Holy Spirit, they determined that only four parts of the law were required; one of those four laws involved not eating blood.
For it has seemed good to the Holy Spirit and to us to lay on you no greater burden than these requirements: that you abstain from what has been sacrificed to idols, and from blood, and from what has been strangled, and from sexual immorality. If you keep yourselves from these, you will do well. (Acts 15:28-29)
There are various views on whether the first three of these four laws are still binding on Christians, even among the authors of this website.
View 1: Some see the first three rules as a concession to the Jews (Acts 15:19-21; perhaps in line with what Paul said in Romans 14 about not putting ‘a stumbling block’ (14:13) in the way of a fellow Christian — see Romans 14:15). Two of these three rules relating to food (perhaps including also the rule about blood) were later relaxed — see Rom.14:3,6,20; 1 Corinthians 8:1-13; and Col.2:16. The transient nature of the Jerusalem Council letter is also highlighted by Jesus’ words about clean foods before Acts 15; e.g. Mark 7:18-19. As a result, many Christians believe the Jerusalem Council second rule forbidding the eating of blood has also been relaxed. The fourth rule in the Jerusalem letter about sexual immorality, however, is reinforced in other parts of the New Testament (e.g. 1 Cor. 6:9-10).
Eat whatever is sold in the meat market without raising any question on the ground of conscience. (1 Corinthians 10:25)
View 2: Others see the rules as still binding, unless explicitly relaxed later. The rules about food sacrificed to idols is relaxed, as explained in Romans 14:3,6,20 and 1 Corinthians 8:1-13. But there are no such instructions about the other rules, and so they should be considered as still binding. So this is something of an argument from silence on blood, given that the first and third rules are relaxed and the fourth is repeated.
In practice today almost all meat sold in butchers is drained of blood, so it is easy to avoid eating blood without having to eat kosher or halal meat. Among common western foods the Jerusalem Council restriction, if still binding on Gentile Christians, would only apply to blood sausage, such as the British ‘black pudding’ or German Blutwurst.