'Bodhi' is Sanskrit for Enlightenment and 'Citta' means Mind. It refers to the wish to attain enlightenment (become a Buddha) for the benefit of all sentient beings. A 'Bodhisattva' is a being (sattva) with the bodhicitta motivation.
A short story:
An enthusiastic student asks his teacher: "Master, what can I do to help all the suffering beings in this world?" The teacher answers: "Indeed, what can you do?"
So, even if I am genuinely concerned about the welfare of others, when I am hopelessly lost in my own problems, trying to deal with the world, how can I help others? I would be like jumping into a river where someone is drowning, when I cannot swim myself...
Therefore, I should first learn to swim myself, learn to deal with my problems, learn how to become liberated from my problems, or at best, become all-knowing or enlightened. The realisation comes: "change the world, start with myself".
This idea is called Bodhicitta: the wish to become an omniscient Buddha so I can be of perfect help for others.
But in order to collect enough positive momentum (Karma) to become a Buddha, I also need to help others as much as possible on my path. But I should realise that at this moment my help is limited, simply because I don't know all the results of my actions.
A short real story as example: one time at Tushita Meditation Center in Dharamsala, India, people who were in a meditation course decided to collect money for the beggars after hearing the benefits of generosity. When looking the next day to hand out the money, only one beggar could be found. The generous people decided to give this beggar all the money. A couple of days later, the beggar was found dead in the street: he had drunk himself to death with all the money.....
Also while helping others, we should not forget the goal of becoming a Buddha to of much more help; therefore ideally, one should be mindful of dedicating any positive energy to this goal.
Some reflections by the Indian saint Shantideva:
Whatever joy there is in this world
All comes from desiring others to be happy,
And whatever suffering there is in this world,
All comes from desiring myself to be happy.
But what need is there to say much more?
The childish work for their own benefit,
The Buddhas work for the benefit of others.
Just look at the difference between them!
Or, as Shantideva reflected the far-reaching thought of Bodhicitta:
May I become food and drink in the aeons of famine for those poverty-stricken suffers.
May I be a doctor, medicine and nurse for all sick beings in the world until everyone is cured.
May I become never-ending wish-fulfilling treasures materialising in front of each of them as all the enjoyments they need.
May I be a guide for those who do not have a guide, a leader for those who journey, a boat for those who want to cross over, and all sorts of ships, bridges, beautiful parks for those who desire them, and light for those who need light.
And may I become beds for those who need a rest, and a servant to all who need servants.
May I also become the basic conditions for all sentient beings, such as earth or even the sky, which is indestructible.
May I always be the living conditions for all sentient beings until all sentient beings are enlightened.
The realisation of Bodhicitta is quite profound, as it is obviously not easy to (automatically) put the welfare of others above one's own welfare. Someone who lives with this realisation is called a Bodhisattva: in all respects a genuine saint.
It may be interesting to note that His Holiness the Dalai Lama considered Mother Theresa a Bodhisattva, and Jesus as well; so Bodhisattvas are not necessarily Buddhists!
"Bodhicitta or the altruistic aspiration to attain Enlightenment for the benefit of all sentient beings is a state of mind which cannot be cultivated or generated within one's mental continuum simply by praying for it to come into being in one's mind. Nor will it come into existence by simply developing the understanding of what that mind is. One must generate that mind within one's mind's continuum.
In order to engage in meditation with sustained effort over a period of time what is crucial is first of all to be convinced of the positive qualities of that mind, and the benefits and merits of generating such a state of mind. It is only when one has seen the qualities, merits and benefits of generating such a state of mind that one will be able to generate within oneself a genuine enthusiasm and perseverance in engaging in a meditation which would enable the individual to generate the mind."
His Holiness the Dalai Lama