You'll want to be using a vector application for the most part. Affinity Designer is a good prosumer option while Inkscape is the opensource option. Sketch is another option but its intended more for Interface design than the two aforementioned but it can be used for logos.
If you mean an app for mobile you could try Adobe Illustrator Draw since you're on iPhone, I believe it exports at least some of the proper logo formats but I don't have an iPhone to confirm or test. You might also just do a search Vector Drawing on iPhone for other options that are in vector format instead of .GIF which is raster. (To understand the difference: What are the differences between vector graphics and raster graphics?)
Based on the image you posted the design should be pretty easy. Draw a rectangle, add text in desired font, use an alignment panel (all 3 of the above applications have this) to center it.
When you're ready to save I'd suggest this question as a good starting point for the correct format: Logo Pack - What should I include?
If you get stuck then come back and create a new question detailing where you got stuck and what you tried.