Things that happen ignominiously occur in a shameful, dishonorable, or underhanded way. If a baseball team is defeated ignominiously, it's so badly trounced that every player feels embarrassed about the game.
If a government uses violence against political protesters, silencing them and blocking their access to reporters and the Internet, it behaves ignominiously. If you cheat on a test, you're acting ignominiously and should feel ashamed about it. In Latin, ignominiosus means "disgraceful or shameful," from the root ignominia, literally "loss of a good name." In other words, if you act ignominiously, you risk losing your good name.