What and where is the connection between 压岁钱 and the character 岁 that appears in it? During the Chinese New Year celebration, one of the various traditions is to give money to younger people in red envelopes. Usually they are given from elderly married couples to younger couples or single people, but it's also common for old people to give them to children.

These red envelopes are called in Chinese 紅包 and 红包 (hóng bāo), while the money is called 壓歲錢 and 压岁钱 (yā suì qián), both first in Traditional and then Simplified characters, respectively. The meaning of this name is literally "the money used to suppress or put down the evil spirit".

But I've met the 岁 character before and that's in the expression used to tell your age:


  我 ... 岁。= I'm ... years old.


And actually, looking up this character, the meaning is "year, years old, (measure word)". 

Why there seems to be no relation with the meaning of the envelopes above? If I look up the expression "压岁钱" in the same dictionary, I get a different meaning from above, which is "money given to children as new year present". 

Does that "year" relates to the meaning above? Where is the link, semantically speaking, between the character and the chinese name for the red envelopes?
"压岁" and "压祟" have the same pronunciation (yā suì). The meaning of "压祟" is exactly what you have mentioned here: "suppress or put down the evil spirit". "压岁钱" is actually a homonym of "压祟钱".  It's a usage passed down from ancient time.