Calderon's popularity has dropped in recent months as the country struggles to recover from an economic recession and a military offensive meant to rein in drug cartels has instead pushed up violence levels.
Al Jazeera's Mariana Sanchez, reporting from Cuidad Juarez, said there were mixed feelings regarding the election.
"There was some voter apathy, but some also felt it was their duty to vote and believe change can be brought about. However, many here also do not trust politicians, considering the security situation has been so bad," she said.
"The governors in many parts of Mexico run their states like fiefdoms, but after this election, we may see them working closely with President Felipe Calderon in light of the deteriorating security situation and the fact that some drug cartels are gaining further ground.
"I think they now know they can no longer work autonomously," our correspondent said.
'Devil they know'
Al Jazeera's Franc Contreras, reporting from Mexico City on Sunday, said people were already talking about the possibility of a 2012 election.
|Tom Ackerman reports on how local elections are a major test of public support for Calderon
"Last year in mid-term elections the PRI took control of the lower house of congress once again," he said.
"Mexicans have not had faith in the right-leaning party that is currently in power here, and they are talking about the devil they know returning to power."
At least three candidates have been assassinated in the run-up to the polls. The murders have been blamed on drug cartels seeking to reinforce their influence.
Mayoral candidates were also killed in Tamaulipas, a northeastern state, and in Chihuahua, a state in the northwest.
The most daring attack so far led to the assassination of Rodolfo Torre, a leading candidate for governor of Tamaulipas, on Monday.
Calderon said the ambush was carried out by "organised crime" - drug cartels looking to sway the vote.
Torre was replaced on the ballot sheet by his brother and voting got under way there at 13:00 GMT on Sunday without incident.
"So far, the process has been totally peaceful in our state, that's why we hope citizens will come out in numbers to vote," Jose Luis Navarro, a Tamaulipas election official, said.
Our correspondent said there is a sense that turnout will be low because people are concerned about drug-gang violence.
"In northern Mexico, people are concerned about drug violence. In the south, the issue is really poverty and questions of what candidates can do to stimulate the economy," Contreras said.
Surge in violence
More than 25,000 people, mainly traffickers and police, have been killed in drug-related violence since Calderon launched an offensive against drug gangs shortly after taking office in 2006.
In the northern state of Sonora, 21 people were killed on Thursday in a gun battle between suspected rival drug gangs.
The previous night, Sandra Salas Garcia, an assistant attorney-general in Chihuahua, was killed by a group of armed men in Ciudad Juarez, one of the cities hit hardest by the surge in drug violence.
Garcia had been working on a special unit charged with investigating the drug trade.
Also in Ciudad Juarez, unidentified men left a severed human head on Thursday outside the house of Hector Murgia, the PRI candidate favoured to win the city's mayoral race.
As voting got under way on Sunday, Pedro Esparza, a factory worker in the border town of Nuevo Laredo, said: "We're a bit nervous, but we have to go out to vote because it's our only weapon for the future."