Hard to explain I will give it a go.
Basically in the NBA there is a salary cap on all the teams this is done to even the playing field and curtail spending so a big market team like the Lakers or Knicks who makes much more than say Memphis or Minnesota are not able to just outspend the other teams and get all the talent. So lets say the salary cap is $58 million that means that all the teams in the league cannot go over paying $58 million in terms of yearly wages or they will pay the penalty. The salary cap is soft (meaning teams can go over it with a hard cap nobody can go over the number which has been stated) so the teams with bigger revenue can and will go over but they pay the price for that because when you do go over you enter the luxury tax. So if the salary cap is at $58 million and you are spending $100 million that means you pay a $100 million while also giving the league $42 million which is how much you are over the cap and the league shares that money among the teams who are not paying the tax. From 13/14 they have changed the rules now and they are even stricter.
If you are $5 million over you pay $1.50 on the dollar so a team that is over by $5 million will pay $7.5 million and repeat offenders at this level will pay $12.5 million
If you are $5-$10 million over you pay $2.50 on the dollar so a team that is over by $10 million will pay $17.5 million and a repeat offender will pay $27.5 million
If you are $10-$15 million over you pay $2.50 on the dollar so a team that is $15 million over will pay $37.5 million and a repeat offender will pay $52.5 million
If you are $15-$20 million over you pay 3.25 on the dollar so a team that is $20 million over will pay $65 million and a repeat offender will pay $85 million
So they have really upped the ante teams like Miami, Knicks, Nets and the Lakers will have to curtail spending or they will pay and that money is shared among non tax paying teams. I don't expect the Nets or Knicks to they are both rich the Lakers may not either.
The cap is soft and teams can go over it they are given some exceptions which they can use to sign players there is the mid-level, rookie, Larry bird exception there are a few other but I don't know them.
Mid-level: the mid level exception can change figure wise and length wise depending on the team and how often they have been over the luxury tax with the heavier repeat offenders getting a lower (MLE) so lets say $5 million for four years for a good team $3.5 million for 3 years for a not too bad offender and $2.5 for two years. After 13/14 the MLE is staying at the stated level and increase by 3% each year.
Rookie: This just means team that teams can go over the salary cap to sign first round draft picks to rookie scale contracts.
Larry Bird: Named after the great player because he was the first player this rule was used for teams are allowed to go over the salary cap to sign their own players at an amount equal to maximum salary. Teams can only use it once so Miami with Wade, Bosh and Lebron can only use that rule on one of them the players has to have played for the same team for 3 seasons also.
The new rules came in under the new collective bargaining agreement made last year which is why the season was shortened because the players and owners struggled to come to an agreement the owners won in the end though
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Hope that helps (had to do some research to make sure my figures where right)