What is the nature of the boundary between the Earth's atmosphere and outer space?

It is not a hard surface, but a gradual diminution of gas density with altitude. For a given input of thermal energy, heavy atoms will travel more sluggishly than light atoms. This means that, in the gravitational field of the Earth, they will also travel to different altitudes. Each atomic and molecular specie has its own 'scale height' which defines over what increment in altitude the density of a given specie decreases by a factor of 2.718 The pressure scale height is about 6 kilometers, so for every 6 kilometers of elevation, the ambient air pressure decreases by a factor of 2.781. Detailed models of the atmosphere give the following parameters for the atmosphere:

altitude(km)...Temperature.......Density(gm/cc)......Particles/cc
0                288 K           0.0012               2.6 x 10^19
10               223             4.1 x 10^-4          1.0 x 10^19
50               273             1 x 10^-6            2.2 x 10^16
100              210             5 x 10^-9            9.5 x 10^12
150              780             2.1 x 10^-12         4.5 x 10^10
300              1500            2.5 x 10^-14         8 x 10^8
1000             1600            1.6 x 10^-18         1.6 x 10^5
.................................................................
At high enough speed, a space craft encounters so many particles per second that even at 1000 km, it feels dynamically like the space craft has encountered a viscous atmosphere. Satellites and the Space Shuttle are measurably affected by the Earth's atmosphere at 150 to 200 kilometers. The outer extent of the atmosphere is also determined by the level of solar activity which heats this ionized plasma and causes it to expand away from the Earth to greater altitudes. Even the Space Station will have to counteract for this effect during its 10 year+ stay in near earth orbit.