When discussing life origins, many theists will say that complexity implies a creator. They will ask how something so complex could happen by accident and use the diversity, variety and complexity of the natural world as a proof for intelligent design.
It seems that humans have pretty much always marveled at nature, at the complexities and diversity of life. The universe is spectacular; from the planets to bacteria, we seek to understand while simultaneously feeling as if we may never fully be able to. Our awe is often coupled with fear, fear of the unknown and fear of the realization that there is so little we can control. This awe and fear has led to the invention of creation myths - stories that attempt to explain how everything got here in the first place and how it continues to be sustained. We have developed many pathways toward understanding; we have learned, and continue to learn, so much.
Theists aren't wrong when they say we still don't have all the answers and that there is so much complexity that science still can't explain. But where they are wrong, is in stating that this complexity necessitates a designer.
Creationists seem to be saying that an even more amazing being must be responsible for the amazing things around us. But if amazing, complex things need a creator, then who created the amazing, complex God? And if God is the exception, then theists are relying on certain "laws" (complex things need a more complex creator) to support their belief, but breaking the laws for those things that don't fit.
It's important to recognize that design and complexity are not the same thing. Looking at something complex and saying it requires design is begging the question and explaining a mystery with a mystery is really no explanation at all.
The theist will also often say:
The human eye is so complex, it clearly serves a distinct purpose; it was designed for something, and so there must be a designer. There's no way that could happen by random chance.
And to a degree, they're correct. Random one-time chance certainly did not form the human eye, but random chance and natural selection aren't the same thing. Their argument betrays a broad misunderstanding of science, and evolution in particular.
Everything that has formed in the natural world, has done so as a result of natural processes following natural laws, repeated over billions of years. This is certainly not "chance," and while it is awe-inspiring, it in no way points to an intelligent designer.
You can find more details on common arguments made by theists and responses to them here.