After the Jewish people had suffered for nearly 2,000 years of exile as clearly foretold by Moses (Lev. 26:38, 44; Deut. 28:64-64) and the Hebrew prophets (Isa. 43:5-6; Jer. 30:11; Joel 3:2; Ezek. 36:8-10; Hos. 9:1-10, etc.), Israel was miraculously reborn as a nation in their ancient homeland on May 14, 1948 (Iyyar 5, 5708). In honor of this historical event, Jews across the world traditionally celebrate Iyyar 5 as Israel's Independence Day, or Yom HaAtzma'ut shel Yisrael (ום העצמאות של ישראל).
The date for Yom Ha'atzmaut can vary from year to year. For instance, it may be moved a day earlier (i.e., to Iyyar 4th) so that it will not conflict with the weekly Sabbath. On our secular calendar Independence Day is therefore observed Tuesday, April 25th at sundown until the following sundown on Wednesday, April 26th.
Note that the word atzma'i (עַצְמָאִי) means "independent" in Hebrew. The word atzma'ut (עַצְמָאוּת) means the state of independence, which comes from atzmi - "my bones" (עֶצֶם). Hence the "Day of Independence" is called Yom Ha'atzmaut in Hebrew. The name reminds us of God's promise to revive the "dry bones" (עֲצָמוֹת) of Israel by bringing the Jewish people back from their long exile (see Ezek. 37:4-5).
עַם יִשְׂרָאֵל חַי / am Yisrael chai: "The people of Israel live!" The nation of Israel is God's "super sign" that He is faithful to His covenant promises (Jer. 31:35-37). Celebrating Israel's independence acknowledges God's loyal love for us all.