There are an estimated 11 million illegal* immigrants in the U.S. Congress funds deportations to the point that about 400,000 get sent back annually, which is the highest rate in modern times. So, at that rate, assuming no new entrants (illegal immigration is estimated to be a net zero currently), we'll get around to deporting all the current illegals in 27.5 years.
* As a legal immigrant, I have no issue with the term "illegal immigrant", and think people who try to change the term are missing the point and not helping. Most entered legally, but they're now here illegally, hence "illegal".
Anyway, a large number of those illegals have legal spouses and/or children born here (and thus citizens). To enforce the rule of law would mean breaking up those families, or deporting legal citizens along with their illegal relatives, which I think is against the constitution. It's a thorny issue that requires Congress to act...
The Senate passed a bill - with bipartisan support, no less - that was an effort to fix this. It's not a great bill, but it's something and that's better than nothing. It is believed that the current House has more than enough votes to pass this bill, and Obama has said he would sign it in a flash. The House leadership has declined to take up the Senate bill and bring it to a vote, has declined to offer a competing bill, and has (lately) threatened to shut down the Government (again) if anyone tries to do anything about this vacuum of their creation.
The executive action taken by Obama is, basically, to shield from deportation about 4 million illegals with legal family ties here, in addition to the 1.5 million currently shielded by DACA. Those to be shielded are required to register themselves, and start paying taxes, or they will still face deportation. Meanwhile, we can carry on deporting the balance of the illegal population, about 5.5 million, and that gives us a window of about 14 years before we have to start slowing down in order to address the names on Obama's protected list. If the House doesn't like that timetable, they can authorize more funding for deportations to speed it up.
Yes, this is a more expanded executive action on immigration than anything undertaken before. But it's on the same principles exactly as executive orders issued by Reagan and Bush Sr. Currently, the problem is larger, hence the scale of the effect of the executive order. The freak-out is entirely political, entirely unjustified and entirely craven. Which is why it's being led by Rafael Cruz.