In large parts of the world, especially Muslim, Hindu, and many African cultures, the right hand is for eating, greeting, giving, and receiving, while the left is traditionally associated with personal hygiene and stays out of social transactions. This is one of the most deeply felt customs a visitor can bump into, precisely because it is about cleanliness rather than mere formality.
In practice: eat with the right hand where hand-eating is the norm, hand over money, gifts, documents, and business cards with the right hand or with both hands, and receive things the same way. In East Asia, both hands is the respectful mode for giving and receiving anything, especially cards and gifts. Left-handed travellers are not expected to relearn writing or lose function, but making the visible social gestures, particularly touching shared food and passing items, with the right hand matters.
If you slip, the consequences are social rather than dramatic: a flicker of discomfort, occasionally a gentle correction. A quick apology and switching hands resolves it completely, and locals readily forgive foreigners who are clearly trying. The mistake to avoid is persisting after you have noticed the discomfort; awareness is the whole game.