Why an Eruv:
The Torah requires that Jews not carry any item, no matter its weight or purpose, in a “Reshus HaRabim” (public domain) on Shabbos. When one carries from house to house and from house to street one does an act of society. When one refrains from carrying on Shabbos one pays tribute to G-d. The law on carrying created severe hardships and diminished the oneg (joy) of Shabbos. The Religious court of King Solomon sought to distinguish between a truly public domain where all carrying is prohibited and a more localized domain, bounded by an eruv where not all carrying is prohibited.
For a more philosophical approach to eruvim, read Leora Lawton’s Drash on Eruvim and Tshuvain the honor of the eruv’s fifth anniversary.
What is an Eruv: The Talmud devotes an entire complicated tractate to Eruvin. An Eruv is an enclosure that legally transforms a series of non-connected private domains (e.g., domain. This domain is not determined by a property deal alone but by its symbolic demarcation, or a system of gates.
What you may do within an Eruv
You May Carry:
Tallit, Chumash, Siddur, or other books; Your house keys; Handkerchief, gloves, pocket watch; Medication; Food to hospital patients; Jackets and other clothing which you remove on warm days; Remove trash from your house if it disturbs Oneg Shabbos; Food from house to Succah (on Shabbat, of course, the law of carrying doesn’t apply on weekday yomim tovim); Reading Glasses
Activities You May Perform:
Push a baby stroller along with food and diapers; Wear a rainhat; Wear Jewelry without concern for Shabbos restrictions; Walk a dog on a leash.
Activities You Still May Not Perform:
Carry items which are Muktseh (may not be touched on Shabbos — e.g., pen); Open or carry an umbrella; Typical weekday activities not in the spirit of Shabbos (e.g. Play ball or ride bike); Swim; Carry anything in preparation for post-Shabbat activity