The Pali word samadhi is literally: 'the (mental) state of being firmly fixed'. It is the focussing of the mind on a single object. It is intense mental application, complete attention. So, concentration practice is exclusive. There is the object of focus - and all else is excluded.
      Let's look at what we mean by 'object'. When we talk about one-pointedness of mind what do we mean by this 'one point?' Mathematically, a point has no dimensions and exists only in relation to its coordinates in space. For practical purposes our object exists contained within specific boundaries. Look at this dot If you look real close you will see it is made of several pixels but we can say the dot is a single object, it has a defined border, clear boundaries. When you are concentrating on the dot you are not looking at the menu on the left of the screen. But... we could define the whole screen as 'an object' - and focus on that - excluding all else. It is important to see the relative nature (size, extent, range) of a meditation object.

I have used the image of the torch beam again (see discussion in 'mindfulness') to try and illuminate the distinction between sati and samadhi. We can generalise and say that mindfulness is more 'broad' with quite loosely defined boundaries. The objects of mindfulness might be multi-faceted and complex whereas the objects used for concentration practice are often singular (I hesitate to say 'simple') with clearly defined boundaries.

Concentration - though sometimes quite weak - is one of the 7 mental factors associated with all consciousness. Samadhi is a specific form of concentration and is similar to sati in that it is directly associated with wholesome mind states. A thief may well have good concentration but their attention is not directed toward freedom from delusion, toward enlightenment.

There are traditionally forty objects used for concentration meditation:
• ten kasinas: e.g. a coloured disc, a piece of earth, a pond
• ten unattractive objects: corpses in different stages of decomposition
• ten recollections: devotional, divinity, death, breath, peace
• four sublime states: see Brahma viharas  § 
• four immaterial states: infinite space + consciousness, nothingness, neither-perception-nor-non-perception.
• one perception: the repulsiveness of food
• one analysis: contemplation of the body in terms of the four primary elements
Generally an object will be chosen to suit one's personality with the unattractive ones being more suitable for lustful types; Brahma viharas for angry types; devotion for the cynical; and so forth. In general you want to choose an object that the mind is going to be able to stay with (big enough to hold) but not so 'big' (interesting, complex, exciting) that you just start thinking about that. It is too simple to say concentration is 'stopping thinking' but thought is a very common hindrance. Put that stick down! See the section on meditation [ § ]

In the text we read of 3 grades of concentraton intensity:
• Preparatory - present at the beginning of one's samadhi practice
• Neighbourhood - i.e. concentration 'approaching' but not yet attaining the 1st absorption (jhana)
• Attainment - i.e. that concentration which is present during the absorptions.
There are many opinions about what degree of concentration is needed. Some say neighbourhood is enough, some say only the jhanas will crack the nut. The main thing is to be putting energy into your practice. The journey is a long one - the details often resolve themselves along the way.

However, the four jhanas make up the usual textual definition of right concentration. So here is what the Buddha says:
"And what, monks, is right concentration? Herein, secluded from sense pleasures, secluded from unwholesome states, a monk enters and dwells in the first jhana, which is accompanied by initial and sustained application of mind and filled with rapture and happiness born of seclusion.
      Then, with the subsiding of initial and sustained application of mind, by gaining inner confidence and mental unification, he enters and dwells in the second jhana, which is free from initial and sustained application but is filled with rapture and happiness born of concentration.
      With the fading out of rapture, he dwells in equanimity, mindful and clearly comprehending; and he experiences in his own person that bliss of which the noble ones say: "Happily lives he who is equanimous and mindful" -- thus he enters and dwells in the third jhana.
      With the abandoning of pleasure and pain and with the previous disappearance of joy and grief, he enters and dwells in the fourth jhana, which has neither-pleasure-nor-pain and purity of mindfulness due to equanimity.
This, monks, is right concentration"
         Digha 22

We can finish by considering what is known as 'moment-by-moment concentration' (khanika-samadhi). The practice here is not to exclude the manifold stream of phenomena but to note any and all phenomenon, clinging to nothing. As this noting develops, concentration becomes stronger until it abides, undistracted, with the constantly changing stream of events. This practice is often found in the Burmese tradition. There is a similar technique known as choiceless awareness.
         Is it sati? Is it samadhi? Concentrate on death, a disc, the breath...? There is much information, many opinions - and what are you to do with it all?

Try this and that. Have fun, stay cool. As long as you are putting effort into the quest all will be well. Of course you will get lost here and there but, hey! that's part of the joy of the journey. Strive on with diligence!