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A common way to do this is to slap a capo on your guitar and play the same notes and chords. Suppose you want to transpose from A to the key of C.
With a capo, you could simply put the capo on the third fret, and then proceed with the same chord formations that are familiar to you in the piece or song in A, which you are wanting to transpose. If you imagine your barre finger (the index finger) as exactly the same as a capo, you immediately understand that you can - at least within reason - use barre chords in place of a capo.
Without a capo, you will have to play a whole new set of chords. For example, if the song is in C and you need to make it a whole step (2 half steps) higher, it would now with now start with D and every other chord would also be played a whole step higher as well. For example: C - Am - Dm - G now becomes D - Bm - Em - A.
We can also use the Circle of Fifths for transposition.