The Colombian exchange had both good and bad effects on both sides of the Atlantic and other areas, primarily in the dispersion of useful plants and animals but also the spread of harmful elements and diseases.
Prior to the Colombian exchange, potatoes and maize (corn) as well as turkeys were unknown outside the Americas. When introduced to other parts of the world, they contributed to a substantially healthier diet. The end result was a healthier population with more children surviving to adulthood, and resulted in an explosion in the population. In the Americas, the introduction of hogs, cattle, sheep, etc. were immensely beneficial to the Europeans who came there, but were generally harmful to Indian settlements as they damaged or destroyed food crops. The most important benefit to Indians was the introduction of the horse by the Spanish. It soon became important in buffalo hunts on the plains. A less welcome arrival in the Americas was the cockroach, which made itself at home and has remained.
Sadly, diseases and other harmful elements were part of the exchange. Thistles and other noxious plants were transported, probably inadvertantly, from Europe to the Americas. Diseases such as small pox, measles and mumps emaciated over 90% of the Indian population. In Europe, the introduction of syphillis from the Americas (previously unknown in Europe but transported there by conquistadores and others from the Americas) decimated that population. Human beings were another sad element of the exchange, as slaves from Africa were introduced into the Americas after attempts to enslave Indians proved unproductive.
Charles C. Mann has an excellent new book, 1493 which provides elaborate detailed information on the Colombian Exchange.
The Columbian Exchange affected the world by mixing things that had been only in the "Old World" with things that had been only in the "New World." For the most part, the Old World gained from this and the New World (especially its native peoples) did not.
For example, the main thing that came from the Old World was disease. The epidemic diseases of the the Old World killed a huge percentage of the natives of the New.
In return, the Old World mostly got things of value. They got, for example, potatoes and tomatoes and tobacco (which we may no longer think is of value...). The Old World also got lots of gold and silver from the New. This meant that the Old World was enriched in a variety of ways by contact with the New.