Congress's primary function - to fund the federal government, is accomplished through the passage of Budgetary Acts that stipulate federal funding numbers and allocate money into each area of spending. This fiscal year, after having failed to repeal the Affordable Care Act in the HoR and having subsequently lost the court battles challenging the law, the GOP are making a desperate attempt at "defunding" the bill by refusing to agree to any budget that includes funds for its implementation. Having submitted several proposals that lack the necessary funding for the program, the GOP are meeting resistance in the dem-controlled Senate, which refuses to pass the proposed budgets. Thus, Congress has failed to budget for the upcoming year, and the government has entered into a partial-shutdown.
It's important to note that no one /intended/ a partisan gridlock-fueled shutdown (this backfired on the GOP in the 1990s and will cause unprecedented economic catastrophe if not rectified by the 16 of October), and that this is merely a consequence of the dysfunction in our legislative branch - which has finally come to a head and culminated in what you see now.
As to the effects: the federal government currently employs 2.1 million people, 800K of which are deemed "non-essential personnel" and will not receive pay-checks. Aside from the fact that this makes mortgage, utility, and insurance payments difficult, a lack of dispensable income means less consumerism.
Investors are also weary, amid the hopeful recovery, another bump in the road has been hit. This is also evidenced by the drop in stock markets across the globe.