Spoiler:

The second ability applies only to casting spells. It does not, for example, change when you may activate abilities that can be activated "only as a sorcery."

Gandalf the White affects a permanent's own enters-the-battlefield and leaves-the-battlefield triggered abilities as well as other triggered abilities that trigger when that permanent enters or leaves the battlefield. Such triggered abilities start with "when" or "whenever."

Replacement effects are unaffected by Gandalf the White's ability. For example, a legendary creature that enters the battlefield with one +1/+1 counter on it won't receive an additional +1/+1 counter.

Abilities that apply "as [this permanent] enters the battlefield," such as choosing a color with Gauntlet of Power, are also unaffected.

You don't need to control the permanent entering or leaving the battlefield, only the permanent that has the triggered ability.

Gandalf the White's effect doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to put counters on a permanent, are also made individually.

If you somehow control two copies of Gandalf the White, a legendary permanent or an artifact entering or leaving the battlefield causes abilities to trigger three times, not four. A third Gandalf the White causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.

If a legendary permanent or an artifact entering or leaving the battlefield at the same time as Gandalf the White (including Gandalf the White itself) causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.

The trigger event doesn't have to specifically refer to legendary permanents or to artifacts, nor does it have to specifically refer to when something "leaves the battlefield." For example, an ability that triggers "whenever a creature dies" would trigger twice if a legendary creature or an artifact creature dies while Gandalf the White is on the battlefield, as would an ability that triggers "whenever a permanent is put into a graveyard from the battlefield."

Look at each legendary permanent and each artifact as it exists on the battlefield, taking into account continuous effects, to determine whether any triggered abilities will trigger multiple times. For example, if a nonlegendary creature that has become legendary (probably because it's a player's Ring-bearer) dies, an ability that triggers when it leaves the battlefield triggers twice.

An ability of a permanent that triggers when a card is put into a graveyard "from anywhere" triggers twice only if Gandalf the White and that permanent are both still on the battlefield immediately after the legendary permanent or the artifact has been put into the graveyard from the battlefield. For example, say you control both Gandalf the White and Sefris of the Hidden Ways. If another legendary creature you control dies, Sefris's first ability will trigger twice. But if Gandalf the White dies, Sefris's first ability will trigger only once.