What does “大概” mean? Are there any words with the same meaning?
I think these other answers are more harmful than they are helpful and here's why: none of them have addressed how 大概 is applied in the real context of using the language. Anyone can look up a word in the dictionary and copy/paste the results here. OP's question is what does 大概 really mean (aka how is it used) and is there anything else like it?

大概 is generally used to mean "approximately", "more or less", or "something like that". Here's an example:

我大概明白你的意思。

What this sentence is saying is: "I don't understand what you're saying completely but I get the gist."

And another:

A：你打了多少个电话？ (How many phone calls did you make?)

B：大概50个。(Approximately 50.)

In terms of other words that capture a similar meaning, the closest approximation I can think of is 左右. Here's how this word could be applied to the example above:

B：50个左右。(50 or thereabouts.)

Here's another example of how using 大概 and 左右 are approximately the same but have a different connotation:

A：你的车买了多少钱？

B1: 大概三十万人民币。(Around 300,000RMB.)

B2: 三十万人民币左右。(300,000RMB more or less.)

In this example, 大概 implies the speaker is unclear or uncertain about the amount, or perhaps they don't remember. The use of 左右 implies a greater level of accuracy or certainty on the part of the speaker but they don't care to go into greater detail. The listener could press the speaker for more detail, in which case they would follow up with a more exact number if they were able.

The thing I rarely if ever hear 大概 used for is to express possibility. What this means is I don't think other words like 可能, 或许, and 也许 are similar in meaning. That said, I'm not a native speaker so I'm willing to allow for a number of use cases I'm not familiar with.