も can mean 'too/also', or 'even', or 'neither/either', depending on if the sentence is affirmative or negative. Multiple meanings like this are quite common for Japanese particles, as there is no need to use different words simply to agree with other statements in the sentence (unlike English).

• 私（わたし）は、先生（せんせい）です。
  I am a teacher.
• 私（わたし）も先生（せんせい）です。
  I'm also a teacher.

As you can see here, English requires 'either/neither (when negative)', or 'too/also', but Japanese only requires も. You can think of it as simply meaning 'Whatever is true for (A), is also true for (B)'.
も is part of many other grammar structures in which it keeps its original meaning, but shifts the nuance a little bit. Some of these include でも, にも～ない, とも. As we can see, it is frequently paired with other particles!

• 車（くるま）は早（はや）い、でも危（あぶ）ない。
  Cars are fast, but dangerous.

Here, も can be seen pairing with で. This suggests that 'With (A), (B) also'. This is regularly translated simply as 'but' in English.