Learn Swedish: Presidenten tar alltid med sig västen. - The president always takes the vest with him.

Sig is a reflexive pronoun that refers back to the subject of the sentence, which is Presidenten (The president). If you used honom, it would mean he is taking the vest with a completely different man.

What does the whole phrase tar med sig mean together?

It translates to "brings" or "takes along." While English often just says "The president brings the vest," Swedish frequently uses the prepositional phrase tar med sig to specifically show that he is transporting the object with himself.

If I want to say "I always take the vest with me," does sig change?

Yes, the reflexive pronoun must match the subject. If the subject is jag (I), the pronoun changes to mig, making the sentence: Jag tar alltid med mig västen.

Why is the word alltid placed right after tar?

In a standard Swedish main clause, the verb is always in the second position. Short adverbs that affect the whole sentence, like alltid (always) or inte (not), generally go directly after that verb, before expressions like med sig.