The Constitution and federal election law stemming from the Civil Rights Movement clearly limit a state's power to deny the right to vote to any U.S. citizen over the age of 18, or to deny voting rates based on race, religion, ethnicity, gender, political persuasion, economic status, or physical/mental capacity. The only class of citizens that individual state are allowed to deny the vote to are felons. Every state has different rules about how long felons can be deprived of the right to vote, following the end of their incarceration. Some states permanently revoke a person's right to vote after that person has been convicted of certain, egregious felonies, while other states provide that a certain period of time must pass without a repeat offense. The federal government has largely left these determinations up to individual states.
That said, although individual states cannot explicitly exclude any class of citizens (aside from felons) from the franchise, those states can implement rules about voting procedure that have an immense impact on suppressing the votes of certain groups, such as the elderly, the economically disadvantaged, and those who do not speak English. The way that states can suppress the votes of people in these categories is by requiring drivers licenses or other state identifications in order to validate a person's identity, or by limiting early and extended voting periods.
Both of these practices make it much harder for those who work long hours and have limited transportation and child care options to vote. Those who cannot afford to have cars often lack drivers licenses, and those who work for low, hourly wages, or cannot afford child care, cannot easily get time off to vote on the day of an election. Low wage workers also tend to move more frequently, so laws that require certain kinds of permanent addresses in order for voters to register, impact those people's ability to vote as well. The same goes for state laws that outlaw same-day registration.
There are numerous other tactics that states can employ to indirectly suppress turnout from groups who may be members of a party or demographic that is a minority in a given state. For instance, many heavily Republican and conservative states in the deep South now require voter ID at polling places, and have banned or gotten rid of extended voting hours, vote by mail, and same day registration. These tactics, which have mostly survived court challenges, tends to lower the turnout of certain voters in the Democratic Party base. Since the state laws that result in this suppression are not explicitly aimed at any group of voters, but are instead justified as measures to prevent voter fraud, appellate courts often uphold them.