As with the definition of Palestine, the definition of Israel varies depending on whom you ask. Whereas I could organize the definitions of Palestine by controversy, every definition of Israel is controversial, and likely to offend at least someone. So, here are the various modern definitions of Israel, from smallest to largest.
Arguably this does not count, but by its smallest definition, Israel does not exist at all with any legitimacy. Extremists like Nasrallah (the leader of Hezbollah), do not recognize Israeli sovereignty within any borders, and view Israel as simply Occupied Palestine. The same view is frequently mentioned by official Palestinian media and other sources.
However, given that a large majority of countries recognize Israel, and given that Israel clearly exercises sovereignty over certain areas, we can reject this definition. Some terrorist groups may not like it, but Israel does, in fact, exist. However, Israel's existence raises another question. Given its existence, what are its borders?
Most countries recognize Israel's sovereignty within the 1949 Armistice Lines.
After Israel declared independence in 1948, the surrounding Arab nations attacked Israel. Israel won, and pushed the armies back. Israel signed agreements with the surrounding Arab states to define the places where the soldiers had stopped fighting, the armistice lines, as temporary de Facto borders.
Every Arab country except Lebanon insisted in the agreements that the armistice lines were temporary, existed only out of military necessity, and should not be construed as final borders. From the armistice agreement with Jordan:
no provision of this Agreement shall in any way prejudice the rights, claims and positions of either Party hereto in the ultimate peaceful settlement of the Palestine question, the provisions of this Agreement being dictated exclusively by military considerations.
The 1949 Armistice Lines acted as borders until 1967, which why they are often referred to as the 1967 borders. Israel's sovereignty is (mostly) recognized within the '49 Armistice Lines, though many countries do not feel that Israel has the right to establish a capital in Jerusalem, even inside the Armistice Lines. However, Israel feels it has rightful sovereignty over a few areas outside the Armistice Lines.
The official Israeli definition of Israel includes Israel within the 1949 Armistice Lines, East Jerusalem, and the Golan Heights.
In 1967, Egypt blockaided the Straits of Tiran, which Israel said it would consider an act of war. In response, Israel destroyed the Egyptian air force. Faced with an invasion from Egypt, Syria, and Jordan, Israel defeated the Arab armies in six days.
In the process, Israel captured the Gaza Strip, the Sinai, the West Bank, and the Golan Heights. Israel gave the Sinai back to Egypt in the 1979 Camp David Accords, and gave the Gaza Strip back as part of the 2005 Unilateral Disengagement Plan. Of the remaining areas, Israel has annexed East Jerusalem and the Golan Heights.
Though Syria rejected Israel's offer to return the Golan Heights in exchange for peace, Syria still claims the Golan Heights for itself, so the international community does not recognize Israel's annexation of the Golan Heights. Though Israel's armistice agreements with Jordan stressed the mutability of borders, the international community does not recognize Israel's annexation of East Jerusalem. As per UN Resolution 242, the UN feels that the final status of these areas should be determined as the result of a peace agreement between Israel and its Arab neighbors.
Though Israel within the Green Line, East Jerusalem, and the Golan Heights are areas that the Israeli government considers part of Israel, Israel controls more land than just that.
If by the term "Israel", one wants to refer to all the land which Israel controls, then this definition of Israel includes Israel within the Armistice Lines, the Golan Heights, East Jerusalem, and Area C of the West Bank.
After Israel captured the West Bank and annexed East Jerusalem, there was a question as to what else to do with the remaining territories. Many Israeli citizens established Settlements in the West Bank, with implicit government approval. However, after a wave of Palestinian rioting in the First Intifada, Israel decided to give Palestine limited self-government in exchange for peace. Israel and the PLO negotiated the Oslo Accords, which granted the Palestinians limited self-government over many areas of the West Bank and Gaza.
The Oslo Accords divided the West Bank and Gaza into Areas A, B, and C. Area A is under full Palestinian civil and military control, and comprises major Palestinian population centers. Area B is under Israeli military control but Palestinian civil control, and Area C is under Israeli civil and military control. Area C comprises the Jordan Valley and all Israeli settlements.
The Oslo Accords were intended to establish only an interim government until a final settlement could be negotiated, but as the Israelis and Palestinians have, as of yet, failed to reach an agreement, the Area divisions in the West Bank still stand (though, as mentioned previously, Israel withdrew from Gaza in 2005). As such, if by "Israel" you mean an area that Israel controls, you are referring to Israel within the Green Line, the Golan Heights, East Jerusalem, and Area C of the West Bank.
Finally, we come to the concept of "Greater Israel". This is not, strictly speaking, a definition of Israel, but since you asked about them, I can talk about it a bit anyway.
Though all previous Israeli peace offers have involved Israel giving up land it controls, there are some elements in the far right-wing that feel that Israel should actually expand. Some, like the now defunct Israeli political party Herut, believe or believed that Israel should expand to comprise the entire Mandate of Palestine.
This derives, in part, from the belief that the Balfour Declaration, which expressed Britain's intent to create a Jewish state in Palestine, promised the Jews a state in all of the Mandate of Palestine. The relevant text of the Balfour Declaration:
His Majesty's government view with favour the establishment in Palestine of a national home for the Jewish people, and will use their best endeavours to facilitate the achievement of this object, it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine, or the rights and political status enjoyed by Jews in any other country.
However, the belief that Israel should include Jordan as well is somewhat of a fringe belief, even among people who read the Balfour declaration in such a way that it promises the Jews Jordan. Given that there are a sufficient number of Arabs just in the West Bank and Gaza to match the number of Jews in Israel, it would be demographically unfeasible for Israel to control Jordan as well.
Israel's long history means that it's hard to settle on a specific definition, and were this answer to go back all the way to the Canaanite s it would be several times longer. However, if you hear a person refer to "Israel" today, and they are at least somewhat educated about the Israeli-Arab conflict, they are likely referring to something like the second or third definitions.
