By oneself: tự … (lấy), một mình

Tự before a verb means you do it yourself, without help: Em tự nấu. I cook for myself. Con tự làm được. I can do it myself. It's about who does the work.

Một mình after the verb means "alone, on your own", with nobody else there: Tôi sống một mình. I live alone. Em thích đi du lịch một mình. I like travelling on my own. It's about company, not help.

Tự also starts some everyday words: tự học (teach yourself, study on your own) and tự giới thiệu (introduce yourself).

In Hà Nội, and in older or written Vietnamese, you'll also meet lấy after the verb, which stresses that nobody helped: tự làm lấy. In Saigon people usually just say tự làm.

Don't confuse một mình (alone) with mình, the friendly word for "I" or "we".