A timing push involves making an attack at a certain point in your build order, such that the attack coincides with the completion of a researched upgrade, or a certain mass of units.
For example: you might see a Terran Bio timing push, where the Terran player moves his forces out such that Stimpack technology will have just finished as they reach the enemy forces.
Also applies to Weapon / Armor upgrades. Since the Weapon / Armor upgrades often completely cancel each other out, it's best to start fights when you have the upgrade, and your opponent does not. If you wait too long, your opponent (if they know what they're doing) will complete their own counter-research, and your advantage will be lost.
Hence, timing push; Timing an attack such that it occurs when you have the momentary advantage.
Like so many things in League of Legends, it depends upon and changes based on your team composition and the enemy's team composition. If you have more poke than they do, just sit outside tower range and throw your poke abilities at them until they're so low that staying near the tower risks dying. This is generally the safest way to force an enemy team off of a tower and then you can kill it uncontested.
Another strat is to instantly clear their wave of minions just outside of tower range and then have your ranged carry get a few hits in every time your minion waves is tanking the tower. However, this requires an ad carry that is very good with positioning and understanding how to not get caught.
Generally you only want to go all in under a tower if you KNOW it will results in good kills for your team. Diving Taric but losing your farmed Vayne is a bad trade, for instance. Other times you would go all in is if someone on their team makes an obvious overextension mistake and because of this you can kill them for "free." Taking down a damage dealer on their team allows your team to engage in more favorable terms - 5v4, even under a tower, is still a pretty good fight to engage on. If you're fighting over a lane tower, teams will usually retreat to avoid the 4v5 fight and defend later.
And, of course, there is the CLG/Hotshot strat: split push. This should generally only be done in a coordinated team setting. Have a champ with teleport, strong pushing, and escape abilities push top(for instance) by himself, and the rest of the team can hover near a tower in bottom lane and force the enemy team to split themselves. When this happens, the person pushing top should teleport in and you can force a 5v4 engage to take the tower.
Importantly, try to never get hit by the tower unnecessarily. If there's not a minion wave there to take the initial agro from the tower, you should just stay out of range. There is almost no chance diving into a more well-equipped team with no minion support will work out in your favor, baron buff or no.
Best Answer
Pushing generally refers to moving the battle lines the minions make towards the opponent's nexus, destroying towers and 'pushing' closer and closer with each wave of minions.
You want to push when:
Both of these situations means you and your minions can attack a tower with impunity, and not worry about being harassed, killed, or countered by enemy heroes.