bueno vs bonito   (good vs cute)
cien vs centavo  (100 vs penny or 100th)
fuerte, fuerza vs fortaleza, fortificación  (strong, force vs fortress, fortification)
hierro vs herramienta (iron vs tool)
huérfano vs orfanato (orphan vs orphanage)
hueso vs osteopatía (bone vs osteopathy--medical study of diseases of bones)
huésped vs hospedería (guest vs hostelry or hospitality)
muela vs molino (mollar vs mill)
mueble vs inmobilario (furniture-moveable vs adjective describing parts of the house not considered furniture because they are immobile)
muerto vs moribundo, morir (dead vs moribund-dying, to die)
muestra vs mostrador (sample vs display case as in a jewelry store)
nueve vs noventa, novecientos (nine vs ninety, nine hundred)
puerta vs portal (door vs portal)
rueda vs rodilla (wheel vs little wheel or knee)
siete vs setenta, setecientos (seven vs seventy, seven hundred)
tiempo vs temporal,  temporario (time vs temporal & temporary)
viejo vs vejita and envejecer (old vs little old lady, to age) 

This pattern of stem changing is one of the key elements that distinguishes Spanish from Portuguese, which does not do this.  There are complex linguistic explanations that can explain which combinations of letters will result in stem changing, so there are predictable patterns.  These rules are not helpful or easy to apply.  The best way to deal with it is to repeat aloud the pairs of words above many times, and do the same thing with a number of stem changing verbs.  Eventually you will develop an intuition as to when a word needs this. I can only describe this by saying that it sort of "gets in your ear."  

As you can see, the process of stem-changing in Spanish is directly related to stressed and unstressed syllables.  Since infinitives are always stressed on the last syllable, the infinitive form does not show the stem change.  Some verb endings place the emphasis on a different syllable than others, so the stem change will appear and disappear depending on the stress. 

The present tense has many stem changing verbs.  The only possible ones are:

e-->ie        o-->ue        e-->i            u-->ue

For example, the verb querer: ro quieres quiere queremos queréis quieren

dormir: mo  duermes  duerme dormimos dormís duermen

jugar:  juego juegas juega jugamos jugáis juegan

pedir:  pido pides pide pedimos pedís piden

For students in SPAN 1100 or higher:

The preterite tense does not have stem changes, except for the IR verbs.  These changes do relate to the rule stated above, but rather to more subtle traits of Spanish phonetics.  In the 3rd persons, the vowel changes from E to I or O to U.

sentir: sentí   sentiste sintió sentimos sentís  sintieron

dormir: dormí  dormiste  durmió  dormimos dormís  durmieron

This stem change also effects the progressive participle and the nosotros form of the present subjunctive.   (sintiendo, durmiendo, sintamos, durmamos)

Present Subjunctive has the same stem changing verbs as the present tense, plus the strange nosotros / vosotros forms of IR verbs.  Remember that most command forms are identical to the Subjunctive.

querer: quiera quieras quiera queramos queráis quieran

dormir: duerma  duermas  duerma durmamos durmáis man

jugar:  juegue juegues juegue juguemos juguéis jueguen

pedir:  pida pidas pida pidamos pidáís pidan

Because of the IR verb problem, I strongly recommend that you learn the IR verbs as a cluster, and learn the progressive participle and the nosotros subjunctive at the same time:

dormir: dormí  dormiste  durmió  dormimos dormisteis  durmieron --duerma  duermas  duerma durmamos durmáis man--durmiendo durmamos

pedir:  pedí   pediste pidió pedimos pedís  pidieron--pida pidas pida pidamos pidáís pidan--pidiendo pidamos

There are no other tenses in Spanish with stem changing verbs.