Star Wars: The Old Republic had posted an explanation of the many ways free players would get only limited access, but an updated chart makes the exact limitations clear.
As a general rule, the restrictions are in threes: three Flashpoints per week, three Warzones per week, three Space Missions per week. Players will also be able to buy weekly passes for these features to get unlimited access. There will also be restrictions on inventory expansions and cargo hold storage for free players, and field revivals on death will be limited. Taking a look at the full chart should make the breakdown obvious (although there's a believed typo under the restrictions on equipping items). Meanwhile, current subscribers can take advantage of a new account feature to see just how many bonus coins they'll get if they're subscribed when the conversion takes place.
[Thanks to Marc for the tip!]