I've seen people use "probable", but others tell me "versxajne" is better. Is there a difference between them? Are there situations when one is correct and the other isn't? If "versxajne" means "probably", then what does "probable" mean?
The traditional word for "probably" is "verŝajne". It is a compound word made from "vera" ('true') and "ŝajni" ('to seem'). So literally it means 'true-seemingly', but from the outset it has only been used in the fixed meaning of 'probably'. Zamenhof formed it in the manner of the German word "wahrscheinlich".
The word "probable" is one of the so-called neologisms, i.e. new word roots that were introduced at some point in order to avoid a compound word that some people disn't like for some reason. It is one of the early neologisms, officialized by the Lingva Komitato (the predecessor of the Akademio de Esperanto) already in 1921, so it was certainly used already before WWI.
Probably the main reasons why it was introduced was that people didn't find the compound "verŝajne" very logical in this meaning, and that there was at the time a general tendency to favour Romance roots over Germanic-based expressions.
"Verŝajne" is still more common, but many people use "probable" instead, or use the two interchangeably. Their meaning is completely identical (at least in common language; below I discuss a possible difference in technical mathematical language). When people say that "verŝajne" is better, this is probably due to a general preference of traditional compounds over neologisms, which I also support (especially because the exaggerated introduction of neologisms is risking to make Esperanto more difficult, and given they are almost all based on European languages, makes the gap in learning difficulty between speakers of European languages and others even bigger).
So I suggest you use the traditional "verŝajne", though "probable" is also correct.
In the mathematical field of statistics, a distinction is made between "probability" and "lilelihood". Maybe a good way to render this distinction in Esperanto is by using "probableco" and "verŝajneco". However the technical vocabulary for statistics is still not fixed enough to say that this distinction is already established. At any rate, this technical distinction doesn't change the fact that the two words are perfect syonyms in common language.
Verŝajne literally means "true-seeming-ly" or verisimilarly and when used in the sense Estas verŝajne, ke... it is equivalent to a confident It looks like... It seems that... The idea is that you are making a judgment based on the information that you have.
Probable, when used in the sense Estas probable, ke... is a literal probably and can be used more generally.
Kredeble is "believably", and when used in the sense Estas kredeble, ke... means It is easy to believe that...
The differences amount to small nuances and not everyone will see them.
Verŝajne, li venis per trajno. It seems very likely that he came by train.
Probable, la toasto alteriĝos sur sian buteritan flankon. The toast will probably land on its buttered side.
Kredeble, tiu kvanto ne sufiĉos. I don't really think that amount will be enough.
Note that on occasion these words retain their role as adverbial forms of verŝajna, probabla, kredebla.
Li ludis sian rolon tre verŝajne. He played his part very realistically.
La ministro malkonsentis, tute probable. The minister disagreed, as was only probable.
Dum sia rakonto, li parolis tre kredeble. While he was talking, his account was very believable.