Iran's government, believe it or not, does have some democratic institutions (whether or not the elections to fill them are reputable is another question). It has a legislature and a president, both selected by Iranian voters. But it also has a supreme leader, whose title makes clear that he's both more powerful than the president and not democratically elected.
The first and perhaps more important take-away from this chart is that the unelected parts of Iran's government are more powerful than the elected parts. An imperfect but useful comparison might be to think of the American political system, in which we elect our president and legislature, arguably the two most powerful branches, but not our supreme court.
Imagine if the U.S. supreme court wasn't just in charge of legal rulings but also directly ran U.S. foreign policy and defense policy, directed powerful agencies from the CIA to the Pentagon to state media to financial institutions, appointed lower judges, led religious institutions and even had the power to remove the president. Now imagine if, instead of nine justice, there were only the chief justice, and he were a religious authority instead of a legal expert. That's a very rough version of the Iranian political system.
The second thing you might notice is the weird overlaps between elected and unelected bodies in the Iranian political system. If it seems like Iran is two totally different systems of government mashed together, a democracy interwoven into a theocracy, that's not an accident. The Islamic Republic was founded by a pair of 1979 referenda, approved by 98 and 99 percent votes, the first of which simply asked, "Islamic Republic: yes or no." As often happens with a referendum in the wake of a popular revolution, people overwhelming supported it, treating it as a less a deliberation on how best to establish their new government than as an endorsement of the revolution. By the time enthusiasm had died down, voters had approved the second referendum, endorsing Ayatollah Ruhollah Khomeini's constitution, which made the country both democratic and theocratic, a contradiction that the country has been struggling with ever since.
Despite the popular perception of Iran as a simple dictatorship, the democratic institutions are still there: legislators fight with one another, presidential candidates spar over major issues, the president and the supreme leader jockey for power. And Iran was founded with some earnestly democratic institutions. But over the past 34 years, perhaps inevitably, power has shifted more and more away from the elected bodies and toward the unelected. It started almost immediately, during Iran's costly eight-year war with Iraq, when Khomeini began consolidating power, as leaders often do during wars. After he died in 1989 and Ali Khamenei became supreme leader, the office he still holds today, he started colluding with the president to push out political rivals. The 2009 election, in which incumbent President Mahmoud Ahmadinejad won reelection in a highly disputed election that Khamenei ultimately backed, was yet another blow to what remains of Iranian democracy.
As another illustration of the supreme leader's overwhelming authority today, which includes the foreign policy and military issues most important to the U.S., here's a chart put together by the U.S. Institute of Peace. It's not as easy to understand as the Post graphic above, but, for the real Iran nerds, it's got a bit more detail:
The president's office is so marginalized that it can be tough to spot, nestled into the bottom-right of the chart. This visualization helps to drive home the supreme leader's breadth of authority, from the bonyads (ostensibly charitable financial institutions linked to the black market, public subsidies and lending markets) to the Islamic Republic of Iran Broadcasting (IRIB, a media office) to the all-important Iranian Revolutionary Guard Corps.
All of this is a handy way to show both the ways that Iran's president matters – he does have real power on domestic policy and its implementation – and the ways that he doesn't, given how much of the country's authorities rest under the supreme leader.