Chair/president voting

The chair votes like any other director

If your bylaws seat the president or chair as a voting member of the board, that seat comes with a full vote on every motion, not a vote that only appears when things are deadlocked. The chair can vote first, vote during a show of hands, or vote on a routine motion to approve last month's minutes, exactly the way any other director can.

"If the president is a member of the voting body, he or she has exactly the same rights and privileges as all other members have."

Source: Robert's Rules of Order Newly Revised, Frequently Asked Questions, Robert's Rules Association

The idea that the chair "stays neutral" unless there's a tie is a common misreading of a much narrower rule about when the chair chooses to use that same vote.

The one real nuance: timing a vote to a tie

Robert's Rules does give the chair a tactical option that isn't quite the same as an ordinary director's: on a non-ballot vote (voice, show of hands, or roll call), the chair can hold back, watch how the count would otherwise land, and then step in and vote specifically to create or break a tie.

"On a tie vote the motion is lost, and the chair, if a member of the assembly, may vote to make it a tie unless the vote is by ballot."

Source: Robert's Rules of Order (1915 edition), constitution.org

This works only because a voice vote or show of hands produces a visible, ongoing count the chair can watch before deciding. It's a right the chair already has as a board member, exercised at the moment where it's decisive, not a second vote stacked on top of an ordinary one. And a tie itself isn't a special case needing a tiebreaker: under a plain majority rule, a tie simply means the motion did not get more than half, so it fails exactly like a vote with more noes than ayes.

Secret ballots remove the option

That timing trick depends on watching the vote happen in real time. A secret ballot doesn't work that way: every member, including the chair, marks and submits a ballot at the same moment, with no running tally to watch. The chair still gets a vote on a ballot question, but the "wait and see" tactic has nothing to attach to when there's no visible count to wait on.

What your bylaws might change

Some associations write their own rule stripping the chair's vote except to break a tie, or removing the tie-vote mechanism altogether. Check your bylaws' section on board officers and voting before assuming the general rule above applies as written. And none of this applies at all unless your bylaws actually adopt Robert's Rules, or another parliamentary authority, as the rulebook for the meetings your board runs.