Metta Bhavana is one of three practises, Sangharakshita ( founder Triratna Buddhist Order ) introduced for people who are new to meditation.
It is a Samatha ( calming ) practise.
Metta means love, warmth, compassion, generosity, patience. I tend to use just the word kind.
Bhavana means to cultivate or bring into being.
So, Metta Bhavana means to cultivate kindness. You are looking to be kind without any expectation of anything in return. Your kindness is unconditional.
By doing the Metta Bhavana practise you cultivate a consciousness that looks for the kind response to a situation. That kind response maybe to challenge someone but you will do so from a place of kindness.
The practise has five stages.
Stage I, Kindness to self.
Stage II, Kindness to a good friend.
Stage III, Kindness to a neutral person. You know they exist but you have no strong emotional response to them.
Stage IV, Kindness to someone you find difficult.
Stage V, Equal kindness to all the people you have put in the practise so far and then include more and more people.
Note that you start with yourself and end with all humanity.
Mindfulness of Breathing and Metta Bhavana work together. You need a base of awareness (mindfulness) to work in any meditation. If some difficult issue comes-up in meditation and it almost undoubtedly will, you will need metta to work on the issue. In Triratna we like to teach all three practises.
Wildmind by Bodhipaksa, Windhorse Publications
Buddhist Mediation by Kamalashila, Windhorse Publications
the Body by Paramananda, Windhorse Publications
the Breath by Vessantara, Windhorse Publications
the Heart by Vessantara, Windhorse Publications
metta by various authors, Windhorse Publications
The Mindful Manifesto by J Heaversedge and E Halliwell, Hay house