Conceit by definition is basically any thought of "I am". In general, some people think of the word conceit as only when people are thinking, "I am great", but no, conceit is anytime when we think of "I am". "I am better than you". "I am not as good as you". "I am equal to you". Those are all levels of conceit, anytime we are thinking of "I, I, I".
Within Buddhism they say that conceit is one of the very last things that disappear for people who get enlightened. So although there are four levels of enlightenment in Buddhism, only the highest level is totally pure, no more aversion, nothing, nothing left, fully enlightened. The level just below that is a level where a person has let go of all sensual desire and all aversion. However, they still have conceit, the thought "I am".
Now, when you are doing the Compassion/Lovingkindness Meditation, the way that we teach it, the phrase, the words that both me and Rosemary use, you will see there is no "I" in there unless, of course, we are doing it for ourselves. But when we are doing it for others there is no "I". The way that I do it is, "May so and so ... find Peace of Mind," there is no "I". When you are wishing Happy Birthday to somebody, you say "Happy Birthday". It's for them that you are wishing Happy Birthday for, and again, there is no "I".
There is a problem for some people doing Compassion/Lovingkindness Meditation in which they don't use a similar form that Rosemary and I teach. Although we do say you can invent your own similar wish, some people change it and they actually start with, "I wish so and so...so and so". That's not it, that's conceit, that's me and them, that's a separation. We want to get rid of that word "I", it's just, "May so and so...find Peace of Mind", etc. That's an area that if you've had that problem with your own wish, then adjust that, get rid of the "I", get rid of the conceit, this "I am doing something".
In reality, yes, we are sitting here and we are doing something. But also in reality, we are not doing anything for other people. We are not actually giving somebody Peace of Mind with this wish. We are changing our mind and heart, we are conditioning ourselves so that we actually have less conceit. So that we actually think more of the well-being of other people rather than just "Me, me" what I want.
As well, in this regard, keeping the phrase exactly the same for everybody is important to let go of conceit because sometimes we may think, "Oh, but, my wife, my husband, my kid, I want to wish them something special". However this changes the level of the unselfish Compassion and Lovingkindness we are trying to develop, whereas keeping the same phrase prevents the conceit of "Me and Mine" to arise.
The more we work on letting go of the "I," then the thought of "Me and Mine" also decreases, then the conceit, "I am", " I have", "This is mine," etc., all of that diminishes, as well. And the more we diminish this "I" the happier we are. It is just that simple.
An example with the Unpleasant Physical Sensations technique. Say you are observing a sensation and you are seeing the size, shape and where it is. Then the minute you think, "MY knee hurts!", there it is, you've got Dukkha. But whenever you are just seeing the sensation and the throbbing and the pressure and this and that, and when there is no "I" there, you have less Dukkha.