Yes. It's like voting for anything. You elect somebody, but you do not control what they do afterward, However, you vote expecting normal behavior by the person that you elected.
For example, when you vote for a state representative or member of congress, you have certain basic expectations: 1) that they will stay a member of their current party; 2) that they will join that party's caucus; 3) that they will support that party's candidates for speaker of the house; 4) that, with some limited exceptions, they will generally support the party's legislative agenda. Sometimes that ends up being wrong and the person changes party, but usually it doesn't.
Similarly, electors have always had the power to vote for whomever they wished. However, the nomination process used by the parties assures that their slate of electors are people who have proven their loyalty to the party in the past. The expectation is that the electors will vote for the party's candidates for president and vice-president. Historically, that expectation has proven accurate.
Over the one hundred fifty year history of the modern system of every state using the popular vote to select electors, there has never been enough defections from the "winning" candidate to send an election into the House of Representatives (and you have to go back to 1836 to find a faithless elector sending the Vice-Presidential race to the Senate). You have to go back to 1972 to find an elector who cast a vote for a nominee of another party, but in all such cases the defection was to a third party candidate.
In the past twelve elections, we have had a grand total of 13 faithless electors, meaning that 99.75% of electors voted as they pledged to vote. So, in practice, my vote and the vote of my state matters at least if the race is close in my state and nationally. On the other hand, if you are a voter in Wyoming or Delaware, your vote probably does not matter in choosing who wins the state and your electors are unlikely to make a difference in who wins.