While Instagram can be great for sharing your own photos and captions, the standard Instagram app doesn't feature the ability to repost other people's pictures. This function can be added, however, through a few apps and Web clients. What options you have will vary based on the smartphone platform you use.
InstaRepost -- Android and iOS
InstaRepost is a free app for both Android and iOS that lets you repost Instagram pictures into your own stream. All reposted images give credit to the original poster through a watermark of their username, although this feature can be removed with an in-app purchase.
Repost for Instagram -- iOS
Repost for Instagram is another free option that adds repost capability to Instagram. With a paid upgrade, you can also like and follow other users from within Repost for Instagram. Repost for Instagram is for iOS only.
Statigram and Webstagram are two Web-based Instagram viewers that offer repost capability. Unlike the repost apps, which offer a direct repost, these sites instead give you the option to email an Instagram picture with the original poster's name on it to your phone. From there you can add it to your own Instagram feed directly.