What does "An adventure for players levels 1-3" mean? I see this on the front of module books. Does this mean the players should begin the adventure between levels 1 and 3, with no hint as to what level the players will end up as at the end of the adventure? Or does it mean that players begin at 1 and end at 3?
Your second statement is correct.
The module expects that the player characters will start at 1st level and finish the module at 3rd level.
Normally the module will provide guidance on how the players should level up, either by following the XP system or by following milestone progression. This information is normally included in the first chapter of the book.
In Curse of Strahd, for example, the "Areas by Level" Table on page 6 demonstrates this by listing the average level players should have when they visit each area. The table's values range from level 1 to 9, which matches the module's "Levels 1-10" rating.
Of course, as the DM, you're free to start the players at any level that you wish. However, especially for low level adventures, the challenges presented may be trivially solvable with higher level spells and abilities, so be sure that you understand the module's challenges, as well as your character's abilities, before trying to scale the level rating yourself.
In the old days, typically it meant that you were supposed to start with characters of that level. For example, Castle Amber (1981) described itself as, "An adventure for Character Levels 3-6" and inside it clarified that it meant "for a party of 6 to 10 characters, between the 3rd and 6th levels of experience. The total of the party’s experience levels should be from 26 to 34, with a total of 30 being best".
In modern times it is more likely to mean that you start at the lower level and end at the higher level. Waterdeep: Dungeon of the Mad Mage (2018) is referred to as being for "Levels: 5 – 20" and explains that this means "taking characters of 5th level or higher all the way to 20th level". While you can start at a level higher than five, this would normally be done by skipping past the easy content and starting at a later point (at least according to this: Does Waterdeep: Dungeon of the Mad Mage allow a party to start after level 6? )
Either can be correct
There is no set guidelines (that I am aware of) regarding the snippets used to advertise a module.
Some modules will include details about leveling up within the adventure. "Lost Mines of Phandelver" is an example 1.
Awarding Experience Points
Divide 200 XP equally among the characters if the party ...
The next stage of the adventure continues with part 3... At some point during part 2, the characters are likely to advance to 3rd level, so make sure the players are keeping track of their XP.
So it is possible that a new character will start at the lower end and be at the higher end when complete.
On the other hand, some modules are designed to be played by various levels of characters, and the encounters are adjusted accordingly.
The basic formula is to add up all the characters levels, then divide by the number of characters, and that becomes the APL (Average Party Level). Then the adventure will make encounters easier or harder depending on the number of characters and that APL.
Too few characters, or too low of an APL, and the encounters are adjusted easier. Too many, or too high and the encounters are adjusted the other way.
An example is "Secrets of Sokol Keep"; which has a lookup table in the beginning to determine if parties are "weak" or "strong". It is labeled as an "adventure for 1st-4th level characters"; meaning that they can be a mix of levels in between those bounds.
Personally, I always think in terms of "what level(s) should the characters start at".
Mostly because, without railroading, there is no way to know for sure how much XP will be awarded by the end of the adventure. After you factor in side quests, creative thinking from PCs, and on-the-fly changes from the DM, it can vary wildly.
1 I know, Lost Mines is advertised as being only for 1st level characters. But it was the only module I had immediate access to that specifically called out about leveling within the adventure.