Within the EU national boundaries are less rigid than within say NAFTA. With the exception of the new accession countries where there are transitional arrangents, any EU citizen has the right of abode and working anywhere in the EU. Many countries are in the Shengen area where there are no border controls between countries at all. Movement between non-Shengen countries for citizens usually involves a minimal examination of either a passport or National Identity document to make sure it is genuine.

By the way absence of passport control is going to be a significant difference between the experience of citixens travelling within the EU and a US citizen travelling outside the country. Not many at the moment realise they will need a passport for re-entry to the USA next year for all countries except Canada and Mexico. These follow a year or so later so that honeymoon trip to Niagra or weekend drinking taquilla is going to be more complicated.

As far as those from outside the EU are concerned, you have to distinguish between legal immigration (for fixed periods like tourists and some seasonal workers or permanently), the (initially) temporary entry of refugees under the Asylum provisions and illegal immigration mostly for economic reasons. Because the level of pay, even for the most menial jobs, is far higher within the EU than in neighbouring countries to the east and south, there is a desire to enter to work even illegally. Such illegal immigrants often forget the additional costs of living but, like the USA are prepared to accept much worse living conditions to enable them to send financial support home.

Freedom of movement within the EU means the external borders have to be "harder" to stop illegal entry. The assimulation eastward means Poland is now has the main border responsibility in the east. Other countries with significant border control problems are Italy which has to tackle speedboats from both Albania and North Africa. Spain has similar problems over the Straights of Gibraltar and als owith its land borders in North Africa. They have recently had an amnsety to regularise illegal entrants.

Illegal economic immigration has often been attmepted in some countries, especially the UK, based on a claim for asylum. The UK has some draconian and quite frankly racist and unpleasant regulations to "tackle" this. There are far more people here illegally as overstayers from temporary visas granted to white Commonwealth countries than fake asylum claimers.

As I alluded to earlier, some people who have been born outside the EU have right of abode as they have a claim to citizenship. These are mostly from the British Commonwealth whose parents are or were UK subjects. Immigration is also granted in the UK for family reunion purposes (so for example a wife and children can join a working father here). You should note that  French Ovrerseas Departements and (I believe) Territories form part of the EU as do the Spanish enclaves in north Africa.

A worker from outside the EU can get a permit to work if they have a skill in shortage - technically a company has to give an assurancve that the post cannot be filled by an EU citizen. A feature of much more of EU immigration compared to the US is the initial intent to return home. In Germany this was formalised as the  "Guestworker" scheme which resulted in many Turks entering to work. Large scale immigration from the Caribbean in the 1950s to the UK was prompted by personnel shortages in public transport and nursing. Very frequently this initial intent to return becomes a more distant ambition although retiring "back home" is getting increasingly popular. Also apparently increasing is investment in the home country to ensure a good income on returement or migration back later in life. Parlicular countries showing this pattern are Ghana and Bangladesh. Some countries like the UK also have schemes to permit agricultural workers from outsie the EU to help at harvesting for a few months. This scheme is becoming redundant as the countries involved acceed to the Union.  

Incidentally, the UK government is currently making proposals that would mean work visas for doctors and nurses are for limited terms and the person will have to return home at the end of the period. This is to stop the attracting of skilled medical workers from the 3rd world and them staying permanantly. The intention of this is to maintain the additional training and experience for the individual at the same time ensuring health provision in the home countries are not as badly disadvantaged.