Default Export (export default)

// MyClass.ts -- using default export
export default class MyClass { /* ... */ }

The main difference is that you can only have one default export per file and you import it like so:

import MyClass from "./MyClass";

You can give it any name you like. For example this works fine:

import MyClassAlias from "./MyClass";

Named Export (export)

// MyClass.ts -- using named exports
export class MyClass { /* ... */ }
export class MyOtherClass { /* ... */ }

When you use a named export, you can have multiple exports per file and you need to import the exports surrounded in braces:

import { MyClass } from "./MyClass";

Note: Adding the braces will fix the error you're describing in your question and the name specified in the braces needs to match the name of the export.

Or say your file exported multiple classes, then you could import both like so:

import { MyClass, MyOtherClass } from "./MyClass";
// use MyClass and MyOtherClass

Or you could give either of them a different name in this file:

import { MyClass, MyOtherClass as MyOtherClassAlias } from "./MyClass";
// use MyClass and MyOtherClassAlias

Or you could import everything that's exported by using * as:

import * as MyClasses from "./MyClass";
// use MyClasses.MyClass and MyClasses.MyOtherClass here

Which to use?

In ES6, default exports are concise because their use case is more common; however, when I am working on code internal to a project in TypeScript, I prefer to use named exports instead of default exports almost all the time because it works very well with code refactoring. For example, if you default export a class and rename that class, it will only rename the class in that file and not any of the other references in other files. With named exports it will rename the class and all the references to that class in all the other files.

It also plays very nicely with barrel files (files that use namespace exports—export *—to export other files). An example of this is shown in the "example" section of this answer.

Note that my opinion on using named exports even when there is only one export is contrary to the TypeScript Handbook—see the "Red Flags" section. I believe this recommendation only applies when you are creating an API for other people to use and the code is not internal to your project. When I'm designing an API for people to use, I'll use a default export so people can do import myLibraryDefaultExport from "my-library-name";. If you disagree with me about doing this, I would love to hear your reasoning.

That said, find what you prefer! You could use one, the other, or both at the same time.

Additional Points

A default export is actually a named export with the name default, so if the file has a default export then you can also import by doing:

import { default as MyClass } from "./MyClass";

And take note these other ways to import exist: 

import MyDefaultExportedClass, { Class1, Class2 } from "./SomeFile";
import MyDefaultExportedClass, * as Classes from "./SomeFile";
import "./SomeFile"; // runs SomeFile.js without importing any exports